%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A lumi.tex TeX Goetz Pfeiffer. %% %% This file contains the TeX manuscript 'Character Values of Iwahori-Hecke %% Algebras of Type B'. %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt]{amsart} \usepackage{euler} %% top matter. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title{Character Values of Iwahori-Hecke Algebras of Type $B$.} \thanks{This article is part of the author's Ph.D.~thesis~\cite{PfDiss} under the direction of Prof.~H.~Pahlings. It is a contribution to the DFG research project ``Algorithmic Number Theory and Algebra''. The author gratefully acknowledges financial support by the DFG and the Studienstiftung des deutschen Volkes.} \author{G\"otz Pfeiffer} \address{Department of Mathematics, University of St Andrews, Fife KY16 9SS, Scotland} \email{goetz@dcs.st-andrews.ac.uk} %% page layout. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \voffset 2.5truepc \hoffset= 2.5truepc \textwidth=5.15in \textheight=8.25in %(incl. running head) \setlength{\evensidemargin}{2.5pc} \setlength{\oddsidemargin}{2.5pc} \setlength{\topmargin}{1pc} \advance\headheight4pt %% paragraph layout. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \parskip 1ex plus 0.5ex minus 0.5ex \parindent 0pt \tolerance=6000 %% use slanted fonts rather than the italic versions. %%%%%%%%%%%%%%%%%%%%% \renewcommand{\itdefault}{sl} %% theorem like environments. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{Thm}{Theorem}[section] \newtheorem{La}[Thm]{Lemma} \newtheorem{Cor}[Thm]{Corollary} \newtheorem{Prop}[Thm]{Proposition} \theoremstyle{definition} \newtheorem{Def}[Thm]{Definition} %% special symbols. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\Q}{\mathbb{Q}} \newcommand{\T}{\mathbb{T}} \newcommand{\Z}{\mathbb{Z}} \let\SS\S \renewcommand{\S}{\mathbb{S}} \newcommand{\ST}{\operatorname{ST}} \newcommand{\Id}{\operatorname{Id}} \newcommand{\Spur}{{\rm trace}} \newcommand{\GAP}{{\sf GAP}} \newcommand{\CHEVIE}{{\sf CHEVIE}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \maketitle The concept of the {\em character table} of a generic Iwahori-Hecke algebra is introduced in~\cite{GePf93} as a square matrix which maps under specialization to the character table of the corresponding Weyl group. The character tables for the series of Iwahori-Hecke algebras of type $A_n$ are determined by a recursion formula which was originally proved in Ram's article~\cite{Ram} and by a different approach in~\cite{Pfeiffer94b}. Apart from $E_8$ the character tables of the Iwahori-Hecke algebras of exceptional type have been computed by Geck in~\cite{GeckHab} and~\cite{Geck94}; for $E_8$ see~\cite{GeMi95}. In this article we determine the character table of the generic Iwahori-Hecke algebra of type $B_n$. The main result is given in Theorem (\ref{thm:chq-B}). This follows from two deformations (\ref{thm:form1}) and (\ref{thm:form2}) of the Murnaghan-Nakayama formula for the character values of symmetric groups. These two results are derived in the more general context of cyclotomic algebras of type $B$ which have recently been defined by Brou{\'e} and Malle~\cite{BrMa93} and by Ariki and Koike~\cite{ArKo}. The central result which allows to derive character formulas of this kind is the decomposition of certain character values in Theorem (\ref{cor:tensor-sum}). The definition of the character table of an Iwahori-Hecke algebra is given in section~\ref{sec:general}. In section~\ref{sec:hck-B} the cyclotomic algebra of type $B$ is introduced; section~\ref{sec:matrix} describes the representing matrices for its irreducible representations. Some facts about wreath products and their characters are collected in sections~\ref{sec:kranz} and~\ref{sec:MNX}. In section~\ref{sec:LR-B} we generalize the Littlewood-Richardson rule for an application to Weyl groups of type $B_n$. The proof of the main results covers sections~\ref{sec:sub} to~\ref{sec:chq-B}. Section~\ref{sec:bem-B} contains some concluding remarks. Computer programs for the explicit calculation of the character tables of generic Iwahori-Hecke algebras of type $A_n$, $B_n$, $D_n$ and their specializations have been implemented in the {\GAP} language~\cite{gap} and are part of {\CHEVIE}~\cite{chevie}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The Character Table of an Iwahori-Hecke Algebra.} \label{sec:general} We briefly recall the definition of the character table of an Iwahori-Hecke algebra from~\cite{GePf93}. Let $(W, S)$ be a Weyl group with generating set of simple reflections $S \subset W$. Let $A$ be the ring of polynomials over $\Z$ in indeterminates $q_s$, $s \in S$, such that $q_s = q_{s'}$ whenever $s$ and $s'$ are conjugate in $W$. The generic Iwahori-Hecke algebra $H$ associated to $W$ is an associative $A$-algebra with basis $T_w$, $w \in W$, and multiplication defined by \[ \begin{array}{ccll} T_{w} T_{w'} & = & T_{ww'} & \mbox{\ if } l(ww')=l(w)+l(w'), \\ T_s^2 & = & q_s T_1 + (q_s - 1) T_s & \mbox{\ for } s \in S, \end{array} \] where $l(w)$ is the usual length function on $W$. Denote by $K$ the algebraic closure of the field of fractions of $A$. Then the $K$-algebra $H_K = K \otimes_A H$ is semisimple and split over $K$. Let $C$ be a conjugacy class of $W$ and denote by $C_{\min}$ the set of elements of minimal length in $C$. Choose an element $w_C \in C_{\min}$ for each class $C$ of $W$. Then we have the following consequence of Theorem (1.1) in~\cite{GePf93}. \begin{Thm} \label{thm:conj} Let $\chi$ be a character of $H_K$. \begin{enumerate} \renewcommand{\labelenumi}{(\alph{enumi})} \item $\chi$ is constant on $\{T_w \mid w \in C_{\min}\}$ for each conjugacy class $C$ of $W$. \item For each $w \in W$ there exist polynomials $f_{w, C} \in \Z[q_s \mid s \in S]$ such that \[ \chi(T_w) = \sum_C f_{w, C}\, \chi(T_{w_C}) \] where the sum is over all conjugacy classes $C$ of $W$. \end{enumerate} \end{Thm} This enables the following natural definition of the character table of $H_K$ which is in fact independent of the actual choice of the $w_C$. \begin{Def} The {\em character table} of $H_K$ is the square matrix $(\phi_i(T_{w_C}))_{i,C}$ with rows labeled by the irreducible characters $\phi_i$ of $H_K$ and columns labeled by the conjugacy classes $C$ of $W$. \end{Def} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Basic Notations.} \label{sec:part} We shortly summarize the most important definitions for this article and cite two important results from the representation theory of symmetric groups. Let $n$ be a positive integer. A {\em partition} $\alpha = [\alpha_1, \dots, \alpha_r]$ of $n$ is a (finite) sequence of integers $\alpha_1 \geq \dots \alpha_r > 0$, $i = 1, \dots, r$, with $\sum_i \alpha_i = n$. We write $|\alpha| = n$ and denote by $l(\alpha) = r$ the {\em length} of the partition $\alpha$. The partition $\alpha$ is represented by a {\em diagram}, which consists of $l(\alpha)$ rows of boxes with $\alpha_i$ boxes in the $i$-th row. This diagram also is denoted by $\alpha$. A {\em tableau} $\T$ assigns a positive integer to each box of the diagram $\alpha$. $\T$ is called {\em semistandard tableau}, if these numbers are increasing along the rows and strictly increasing along the columns. If $\beta_i$ is the number of entries $i$ in $\T$, then $\beta = [\beta_1, \beta_2, \dots]$ is called {\em content} of $\T$ and $\alpha$ is the {\em shape} of $\T$. A semistandard tableau with content $[1^n] = [1, 1, 1, \dots]$ is called {\em standard tableau}. In a standard tableau each of the numbers $1, \dots, n$ occurs exactly once. The following picture shows from left to right the diagram of the partition $\alpha = [4, 2, 1]$ of $7$, the {\em canonical} standard tableau of shape $\alpha$, a semistandard tableau of shape $\alpha$ with content $\beta = [3, 2, 1, 1]$ and a tableau of shape $\alpha$, which is not semistandard. \begin{center} \setlength{\unitlength}{0.01in}% \begin{picture}(440,60)(40,740) \put( 40,800){\line( 0,-1){ 60}} \put( 40,740){\line( 1, 0){ 20}} \put( 60,740){\line( 0, 1){ 60}} \put( 60,800){\line( 1, 0){ 60}} \put(120,800){\line( 0,-1){ 20}} \put(120,780){\line(-1, 0){ 80}} \put( 40,780){\line( 0, 1){ 20}} \put( 40,760){\line( 1, 0){ 40}} \put( 80,760){\line( 0, 1){ 40}} \put( 40,800){\line( 1, 0){ 20}} \put(100,800){\line( 0,-1){ 20}} \put(160,800){\line( 0,-1){ 60}} \put(160,740){\line( 1, 0){ 20}} \put(180,740){\line( 0, 1){ 60}} \put(180,800){\line( 1, 0){ 60}} \put(240,800){\line( 0,-1){ 20}} \put(240,780){\line(-1, 0){ 80}} \put(160,780){\line( 0, 1){ 20}} \put(160,760){\line( 1, 0){ 40}} \put(200,760){\line( 0, 1){ 40}} \put(160,800){\line( 1, 0){ 20}} \put(220,800){\line( 0,-1){ 20}} \put(280,800){\line( 0,-1){ 60}} \put(280,740){\line( 1, 0){ 20}} \put(300,740){\line( 0, 1){ 60}} \put(300,800){\line( 1, 0){ 60}} \put(360,800){\line( 0,-1){ 20}} \put(360,780){\line(-1, 0){ 80}} \put(280,780){\line( 0, 1){ 20}} \put(280,760){\line( 1, 0){ 40}} \put(320,760){\line( 0, 1){ 40}} \put(280,800){\line( 1, 0){ 20}} \put(340,800){\line( 0,-1){ 20}} \put(400,800){\line( 0,-1){ 60}} \put(400,740){\line( 1, 0){ 20}} \put(420,740){\line( 0, 1){ 60}} \put(420,800){\line( 1, 0){ 60}} \put(480,800){\line( 0,-1){ 20}} \put(480,780){\line(-1, 0){ 80}} \put(400,780){\line( 0, 1){ 20}} \put(400,760){\line( 1, 0){ 40}} \put(440,760){\line( 0, 1){ 40}} \put(400,800){\line( 1, 0){ 20}} \put(460,800){\line( 0,-1){ 20}} \put(165,785){$1$} \put(185,785){$2$} \put(205,785){$3$} \put(225,785){$4$} \put(165,765){$5$} \put(185,765){$6$} \put(165,745){$7$} \put(285,785){$1$} \put(305,785){$1$} \put(325,785){$1$} \put(285,765){$2$} \put(345,785){$2$} \put(285,745){$3$} \put(305,765){$4$} \put(405,785){$1$} \put(405,765){$1$} \put(405,745){$1$} \put(425,765){$3$} \put(425,785){$2$} \put(445,785){$3$} \put(465,785){$4$} \end{picture} \end{center} Let $\T$ be a standard tableau of shape $\alpha$. Each number $m \in \{1, \dots, n\}$ has a unique position in $\T$. If $m$ has position $(i,j)$ in $\T$, i.~e.~$m$ lies in the $i$-th row and in the $j$-th column, then we denote by $c(\T:m) = j - i$ its {\em content}. This way the content measures the distance of $m$ to the diagonal of the tableau $\T$. Furthermore the {\em axial distance} $r(m_1, m_2)$ between $m_1$ and $m_2$ in $\T$ is defined as $r(m_1, m_2) = c(\T:m_2) - c(\T:m_1)$. The content $c(\alpha)$ of the partition $\alpha$ is defined by $c(\alpha) = \sum_{m=1}^{n} c(\T:m)$ and it is independent of the chosen standard tableau of shape $\alpha$. In the above canonical tableau of shape $\alpha = [4, 2, 1]$ the numbers $1$ and $6$ have content $0$, the number $5$ has content $-1$ and the number $2$ has content $1$. The axial distance $r(2, 5)$ therefore is $-1 - 1 = -2$. The content of the partition $\alpha$ is $c(\alpha) = 3$. For partitions $\alpha$ and $\gamma$ with $l(\alpha) \leq l(\gamma)$ and $\alpha_i \leq \gamma_i$ for $i = 1, \dots, l(\alpha)$ we write $\alpha \subseteq \gamma$. Let $\alpha$ and $\gamma$ be partitions with $\alpha \subseteq \gamma$. Then the set theoretic difference $\gamma \setminus \alpha$ of the diagrams $\gamma$ and $\alpha$ forms a {\em skew diagram}. The diagram $\gamma \setminus \alpha$ is a {\em strip}, if it does not contain any $2 \times 2$-block of boxes. If $\gamma \setminus \alpha$ is furthermore connected then it is called a {\em hook}. The {\em hook length} of a hook $\gamma \setminus \alpha$ is denoted by $l^{\gamma}_{\alpha}$ and is defined as the number of rows which are occupied by the hook in the diagram $\gamma$, minus one. The connected components of a strip $\gamma \setminus \alpha$ are hooks and we denote by $c^{\gamma}_{\alpha}$ their number and by $l^{\gamma}_{\alpha}$ the sum of the corresponding hook lengths. The conjugacy classes and the irreducible characters of the symmetric group $\S_n$ are parameterized by the partitions of $n$. Its character table is determined by the Murnaghan-Nakayama formula, which can now be stated. For this let $\chi^{[\ ]} = 1$. \begin{Thm} [Murnaghan-Nakayama formula] \label{thm:MN} Let $\gamma$ and $\pi$ be partitions of $n$ and let $k = \pi_i$ for some $i \leq l(\pi)$. Furthermore let $\rho = [\pi_1, \dots, \pi_{i-1}, \pi_{i+1}, \dots, \pi_{l(\pi)}]$. Then \[ \chi^{\gamma}(\pi) = \sum_{|\gamma \setminus \alpha| = k} (-1)^{l^{\gamma}_{\alpha}} \chi^{\alpha}(\rho), \] where the sum is taken over all partitions $\alpha$ of $n-k$ such that $\gamma \setminus \alpha$ is a hook. \end{Thm} \begin{proof} \cite{JaKe} (2.4.7), \cite{Kerber91} (5.6). \end{proof} Recall that a {\em lattice permutation} is a sequence of positive integers where, for all $i,j$, the number of integers $i$ among the first $j$ elements is greater than or equal to the number of integers $i+1$ among these. Let $\gamma$ be a partition of $n$, $\alpha$ a partition of $n-k$, $\beta$ a partition of $k$ and denote by $g^{\gamma}_{\alpha \beta}$ the multiplicity of the character $\chi^{\alpha} \times \chi^{\beta}$ in the restriction of the character $\chi^{\gamma}$ of $\S_n$ to $\S_{n-k} \times \S_k$. \begin{Thm}[Littlewood-Richardson rule] \label{thm:LR} For partitions $\alpha$ of $n-k$, $\beta$ of $k$ and $\gamma$ of $n$ the multiplicity $g^{\gamma}_{\alpha \beta}$ equals the number of skew semistandard tableaux of shape $\gamma \setminus \alpha$ with content $\beta$, which yield lattice permutations if the entries are read row by row from right to left. In particular $g^{\gamma}_{\alpha\beta} = 0$, whenever $\alpha \not\subseteq \gamma$. \end{Thm} \begin{proof} \cite{JaKe}, (2.8.13). \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The Cyclotomic Algebra of Type $B_n^{(r)}$.} \label{sec:hck-B} This section introduces the algebras we are dealing with. Let $r$ be a positive integer. Following~\cite{ArKo} and~\cite{BrMa93} we define a cyclotomic algebra associated to the complex reflection group $C_r \wr \S_n$ (s. sections~\ref{sec:kranz} and~\ref{sec:MNX} for some facts about groups of this type). For this let $q, Q_0, \dots, Q_{r-1}$ be indeterminates over $\Q$. Furthermore let $A = \Z[q, Q_0, \dots, Q_{r-1}]$ and let $K$ be the field of fractions of $A$. \begin{Def} \label{def:hck-B} The {\em cyclotomic algebra} $H_n^{(r)}$ of type $B^{(r)}_n$ is generated as $K$-algebra by elements $T, S_1, \dots, S_{n-1}$ satisfying the relations \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} \item $(T - Q_0) \dots (T - Q_{r-1}) = 0$, \item $S_i^2 = q + (q - 1)S_i$ \quad for $i = 1, \dots, n-1$, \item $T S_1 T S_1 = S_1 T S_1 T$, \item $T S_i = S_i T$ \quad for $i > 1$, \item $S_i S_j = S_j S_i$ \quad for all $i,j$ with $|i - j| \geq 2$, and \item $S_i S_{i+1} S_i = S_{i+1} S_i S_{i+1}$ \quad for $i = 1, \dots, n-2$. \end{enumerate} \end{Def} The relations (iii) -- (vi) are called {\em braid relations}. \begin{Prop} $H_n^{(r)}$ is a semisimple algebra of dimension $n!\ r^n$ over $K$ and it splits over $K$. \end{Prop} \begin{proof} \cite{ArKo} (3.10). \end{proof} The elements $S_1, \dots, S_{n-1}$ generate a subalgebra of type $A_{n-1}$ which will be denoted by $H_n$ for short. We define elements $T_i \in H^{(r)}_n$ as $T_0 = T$ and $T_i = S_i T_{i-1} S_i$ for $i = 1, \dots, n-1$. Then (\cite{ArKo}, (3.4)) \[ H^{(r)}_n = \sum_{\substack{w \in \S_n \\ 0 \leq k_i \leq r-1}} K T_0^{k_0} \dots T_{n-1}^{k_{n-1}} T_w \] as a $K$-vector space. By specializing $q \mapsto 1$ and $Q_i \mapsto \zeta^i$ for a primitive $r$-th root of unity $\zeta$ one obtains from $H^{(r)}_n$ the group algebra of the wreath product $C_r \wr \S_n$. This specialization gives rise to a bijection of the irreducible characters by Tits' Deformation Theorem (\cite{CR}, (68.17)). For $r = 2$ and $Q_1 = -1$ the algebra $H^{(2)}_n$ is the Iwahori-Hecke algebra of type $B_n$ with parameters $q_{s_i} = q$, $i = 1, \dots, n-1$, and $q_t = Q_0$. In this case we denote $Q_0$ simply by $Q$. From section~\ref{sec:res-BA} on we restrict our investigations to this case. Finally note that for $r = 1$ and $Q_0 = 1$ the algebra $H^{(r)}_n$ is the Iwahori-Hecke algebra $H_n$ of type $A_{n-1}$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Tuples of Partitions and Further Notation.} \label{sec:ptupel} We have to generalize some of the notation from section~\ref{sec:part} before we proceed. An {\em $r$-tuple of partitions of $n$} is a sequence $\gamma = (\gamma^0, \dots, \gamma^{r-1})$ of $r$ partitions with $\sum_{i = 0}^{r-1} |\gamma^i| = n$. A $2$-tuple of partitions is also called a {\em double partition}. The $r$-tuple of partitions $\gamma$ is represented by the $r$-tuple of diagrams $\gamma^i$, $i = 0, \dots, r-1$ . This diagram also is denoted by $\gamma$. If $\alpha$ and $\gamma$ are $r$-tuples of partitions with $\alpha^i \subseteq \gamma^i$ for $i = 0, \dots, r-1$, then we write $\alpha \subseteq \gamma$. In this case the set theoretic difference $\gamma \setminus \alpha$ is a {\em skew diagram}. A {\em strip} $\gamma \setminus \alpha$ does not contain any $2 \times 2$-blocks, and a {\em hook} is a connected strip. For a hook $\gamma \setminus \alpha$ we have $\alpha^i = \gamma^i$ for all $0 \leq i \leq r-1$ except $i = \tau(\gamma\setminus\alpha)$. This means, $\tau(\gamma\setminus\alpha)$ denotes the position of the component of $\gamma$, which (completely) contains the hook $\gamma \setminus \alpha$. A {\em tableau} $\T$ of {\em shape} $\gamma$ is an $r$-tuple $(\T^0, \dots, \T^{r-1})$ of ordinary tableaux $\T^i$, where $\T^i$ is of shape $\gamma^i$ for $i = 0, \dots, r-1$. Such a tableau is called {\em semistandard tableau}, if each component $\T^i$ is an (ordinary) semistandard tableau, and it is called {\em standard tableau}, if it furthermore contains each of the numbers $1, \dots, n$ exactly once. Let $k \leq n$ and let $\alpha$ be an $r$-tuple of partitions of $n-k$ with $\alpha \subseteq \gamma$. A (skew) {\em standard tableau} $\T$ of shape $\gamma \setminus \alpha$ maps each box of the diagram $\gamma \setminus \alpha$ to one of the numbers $n-k+1, \dots, n$, such that these numbers strictly increase along the rows and along the columns of each component of $\T$. The number of standard tableaux of shape $\gamma$ (resp.\ $\gamma \setminus \alpha$) is denoted by $f^{\gamma}$ (resp. $f^{\gamma \setminus \alpha}$). Denote by $\ST(\gamma)$ the set of all standard tableaux of shape $\gamma$. \begin{La} \label{la:st-bij} Let $k \leq n$ and let $\gamma$ be an $r$-tuple of partitions of $n$. The map \begin{align*} \ST(\gamma) & \rightarrow \bigcup_{|\gamma \setminus \alpha| = k} \ST(\alpha) \times \ST(\gamma \setminus \alpha) \\ \T & \mapsto (\T_1, \T_2), \end{align*} which decomposes each standard tableau $\T$ of shape $\gamma$ into a part $\T_1$, which contains the numbers $1, \dots, n-k$, and a part $\T_2$, which contains the numbers $n-k+1, \dots, n$, is a bijection. \end{La} \begin{proof} Since $\T$ is a standard tableau, the boxes of $\T$ which contain the numbers $1, \dots, n-k$ form a diagram $\alpha \subseteq \gamma$ for some $r$-tuple of partitions $\alpha$ of $n-k$. Conversely, a standard tableau of shape $\alpha$ and a standard tableau of shape $\gamma \setminus \alpha$ are uniquely combined to a standard tableau of shape $\gamma$. \end{proof} If $\T$ is a standard tableau, define $\tau(\T:m)$ for $m \leq n$ such that the number $m$ is contained in the component $\T^{\tau(\T:m)}$ of $\T$. The {\em content} of $m$ in $\T$ is defined to be $c(\T:m) = i - j$ if $m$ lies in the $i$-th row and in the $j$-th column of this component. The {\em axial distance} between $m_1$ and $m_2$ in $\T$ is defined as $r(m_1, m_2) = c(\T:m_2) - c(\T:m_1)$. The symmetric group $\S_n$ acts on the set of all tableaux of shape $\gamma$ and of content $[1^n]$ by permuting the entries. So the image $\T (i, i+1)$ of $\T$ under $s_i = (i, i+1)$ arises from $\T$ by exchanging the entries $i$ and $i+1$ in $\T$. Here the image $\T (i, i+1)$ of a standard tableau $\T$ is a standard tableau, if and only if the numbers $i$ and $i+1$ are neither contained in the same row nor in the same column of a component of $\T$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Wreath Products and the Weyl Group of Type $B_n$.} \label{sec:kranz} Let $X_n = C_r \wr \S_n$ be the wreath product of the cyclic group $C_r$ of order $r$ and the symmetric group $\S_n$ on $n$ points. The elements of $X_n$ are of the form $x = (f; \sigma) = (f_1, \dots, f_n; \sigma)$ with $f_i \in C_r$, $i = 1, \dots, n$ and $\sigma \in \S_n$. For an element $x = (f; \sigma) \in X_n$ and a $k$-cycle $\kappa = (j, j\kappa, \dots, j\kappa^{k-1})$ of $\sigma$ the product \[ g((f; \sigma), \kappa)\:= f_j f_{j\kappa} \dots f_{j\kappa^{k-1}} \] is an element of $C_r = \left$ and is called {\em cycle product} of $x$ and $\kappa$. The {\em cycle structure} of $x$ is an $r$-tuple of partitions $\pi$ which is defined as follows. For this let $\pi^j$, $j = 0, \dots, r-1$, be the partition which contains a part $k$ for each $k$-cycle $\kappa$ of $\sigma$ with $g((f; \sigma), \kappa) = c^j$. Then $\pi = (\pi^0, \dots, \pi^{r-1})$ is an $r$-tuple of partitions and the conjugacy classes of $X_n$ are described in the same way as the conjugacy classes of the symmetric group. \begin{Prop} Two elements of $X_n$ are conjugate if and only if the have the same cycle structure. \end{Prop} \begin{proof} \cite{JaKe} (4.2.8). \end{proof} Thus the conjugacy classes of $C_r \wr \S_n$ are parameterized by the $r$-tuples of partitions of $n$. The Weyl group $W_n = W(B_n)$ of type $B_n$ is generated by $S = \{t, s_1, \dots, s_{n-1}\}$ subject to the relations implied by the following Dynkin diagram. \begin{center} \unitlength.015in \begin{picture}(200,30) \put( 30, 10){\circle*{5}} \put( 30, 8){\line(1,0){30}} \put( 30, 12){\line(1,0){30}} \put( 60, 10){\circle*{5}} \put( 60, 10){\line(1,0){30}} \put( 90, 10){\circle*{5}} \put( 90, 10){\line(1,0){20}} \put(120, 10){\circle*{1}} \put(130, 10){\circle*{1}} \put(140, 10){\circle*{1}} \put(150, 10){\line(1,0){20}} \put(170, 10){\circle*{5}} \put( 27, 20){$t$} \put( 57, 20){$s_1$} \put( 87, 20){$s_2$} \put(160, 20){$s_{n-1}$} \end{picture} \end{center} The elements $s_1, \dots, s_{n-1}$ generate a subgroup of $W_n$ which is isomorphic to $\S_n$. The {\em sign changes} $t_i$ defined by $t_0 = t$ and $t_i = s_i t_{i-1} s_i$ for $i = 1, \dots, n-1$, generate an elementary abelian normal subgroup of order $2^n$ in $W_n$. For each $k \leq n$ the subgroup $\left$ of $W_n$ is isomorphic to $W_k$ and the subgroup $\left$ is isomorphic to $W_{n-k} \times W_k$. $W_n$ is isomorphic to the wreath product $C_2 \wr \S_n$. Hence the general results for wreath products apply to this group. In particular the conjugacy classes of $W_n$ are parameterized by the double partitions of $n$. A standard representative of minimal length in the conjugacy class of $W_n$ which belongs to the double partition $\pi = (\pi^0, \pi^1)$ of $n$ is constructed as follows. Let $l = l(\pi^1)$ and define a sequence $\sigma$ of sums by \[ \sigma_i = \begin{cases} 0 & \text{for $i = 0$,} \\ \sigma_{i-1} + \pi^1_{l-i+1} & \text{for $i = 1, \dots, l$, and} \\ \sigma_{i-1} + \pi^0_{i-l} & \text{for $i = l+1, \dots l + l(\pi^0)$.} \end{cases} \] The representative arises from the word $s_0 s_1 \dots s_{n-1} s_n$ by replacing $s_i$ by $t_i$ for all $i \in \sigma$ such that $i < \sigma_l$ and by omitting $s_i$ for all remaining $i \in \sigma$. Note that always $0$ and $n$ are in $\sigma$, whence the undefined symbols $s_0$ and $s_n$ are always replaced or omitted. If, for instance, $\pi = ([3], [4, 1])$ is a double partition of $8$, then $\sigma = (0, 1, 5, 8)$ with $l = 2$. This yields $t_0 t_1 s_2 s_3 s_4 s_6 s_7 = t s_1 t s_1 s_2 s_3 s_4 s_6 s_7$ as a representative of minimal length for this class. A presentation of the wreath product $C_r \wr \S_n$ is obtained by specializing $q \mapsto 1$ and $Q_i \mapsto \zeta^i$ for a primitive $r$-th root of unity $\zeta$ in the Definition~\ref{def:hck-B} of the cyclotomic algebra $H_n^{(r)}$. We get the same presentation by replacing the relation $t^2 = 1$ by the relation $t^r = 1$ in the above presentation of $W(B_n)$ by the Dynkin diagram. The group $C_r \wr \S_n$ is a {\em complex reflection group}, i.~e., it has a complex representation in which the generators are {\em pseudo reflections} ($n \times n$-matrices with $n-1$ eigenvalues 1). In the classification of the irreducible finite complex reflection groups by Shephard and Todd~\cite{ShTo} $C_r \wr \S_n$ is denoted by $G(r, 1, n)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Characters of Wreath Products.} \label{sec:MNX} The irreducible characters of $X_n = C_r \wr \S_n$ are parameterized by the $r$-tuples of partitions of $n$. The character $\chi^{\alpha}$ corresponding to an $r$-tuple $\alpha = (\alpha^0, \dots, \alpha^{r-1})$ of partitions of $n$ can be described as follows (cf.~\cite{JaKe} chapter 4). For this let $\phi_0, \dots, \phi_{r-1}$ be the irreducible characters of $C_r$. The outer tensor product $\prod_{c=0}^{r-1} \phi_c^{|\alpha^c|}$ of $|\alpha^c|$ copies of $\phi_c$ for each $c = 0, \dots, {r-1}$, is an irreducible character of the normal subgroup $C_r^n$ of $X_n$. It can be extended in a natural way to the character \[ \prod_{c=0}^{r-1} \overline{\phi_c^{|\alpha^c|}} \] of its inertia group \[ T_{\alpha} = T_{\alpha^0} \times \cdots \times T_{\alpha^{r-1}} \cong X_{|\alpha^0|} \times \cdots \times X_{|\alpha^{r-1}|} . \] Tensoring this character with the irreducible character $\prod_{c=0}^{r-1} \chi^{\alpha^c}$ of the inertia factor \[ T_{\alpha}/C_r^n \cong \S_{|\alpha^0|} \times \cdots \times \S_{|\alpha^{r-1}|} \] and inducing to $X_n$ yields the character $\chi^{\alpha}$ of $X_n$. Conversely each irreducible character of $X_n$ has this form \[ \chi^{\alpha} = \Bigl(\prod_{c=0}^{r-1} \overline{\phi_c^{|\alpha^c|}} \otimes \chi^{\alpha^c} \Bigr)^{X_n}. \] \begin{Prop} The characters $\chi^{\alpha}$ form a complete system of pairwise different absolutely irreducible characters of $X_n$ if $\alpha$ runs through all $r$-tuples of partitions of $n$. \end{Prop} \begin{proof} \cite{JaKe} (4.4.3). \end{proof} The character table of $C_r \wr \S_n$ is determined by the following generalization of the Murnaghan-Nakayama formula (\ref{thm:MN}). \begin{Prop} \label{cor:MNX} Let $\zeta$ be a primitive $r$-th root of unity, let $\gamma$ and $\pi$ be $r$-tuples of partitions of $n$, and let $k = \pi^t_i$ for a $t < r$, $i \leq l(\pi^t)$. Moreover let $\rho$ be the $r$-tuple of partitions of $n-k$, which is obtained from $\pi$ by deleting the part $\pi^t_i = k$. The value of the irreducible character $\chi^{\gamma}$ on an element $x \in C_r \wr S_n$ with cycle structure $\pi$ is determined by \[ \chi^{\gamma}(\pi) = \sum_{|\gamma \setminus \alpha| = k} \zeta^{st} (-1)^{l_{\alpha}^{\gamma}} \chi^{\alpha}(\rho), \] where the sum ranges over all $r$-tuples of partitions $\alpha$ of $n-k$ such that $\gamma \setminus \alpha$ is a hook and where $s = \tau(\gamma \setminus \alpha)$. \end{Prop} \begin{proof} This follows from a more general formula for wreath products of finite groups with symmetric groups (s.~\cite{Pfeiffer94a} (4.4) and~\cite{Stembridge} (4.3)) by the fact that $(\zeta^{st})$ is the character table of the cyclic group $C_r$. For an explicit proof see~\cite{ArKo} (2.2). \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Two Littlewood-Richardson Rules.} \label{sec:LR-B} In this section we derive two generalizations of the Littlewood-Richard\-son rule for Weyl groups $W_n$ of type $B_n$. For this let $\alpha$, $\beta$ and $\gamma$ be double partitions of $n-k$, $k$ and $n$ and let $g^{\gamma}_{\alpha \beta}$ be the multiplicity of the character $\chi^{\alpha} \times \chi^{\beta}$ in the restriction of the character $\chi^{\gamma}$ of $W_n$ to $W_{n-k} \times W_k$. Then the computation of this multiplicity can be reduced to the classical Littlewood-Richardson rule (\ref{thm:LR}). \begin{Prop} \label{thm:LR-B} For double partitions $\alpha =(\alpha^0, \alpha^1)$ of $n-k$, $\beta = (\beta^0, \beta^1)$ of $k$ and $\gamma = (\gamma^0, \gamma^1)$ of $n$ we have $g^{\gamma}_{\alpha \beta} = g^{\gamma^0}_{\alpha^0 \beta^0} g^{\gamma^1}_{\alpha^1 \beta^1}$. \end{Prop} \begin{proof} By Frobenius reciprocity $g^{\gamma}_{\alpha \beta}$ also is the multiplicity of $\chi^{\gamma}$ in the induced character $(\chi^{\alpha} \times \chi^{\beta})^{W_n}$, hence \[ (\chi^{\alpha} \times \chi^{\beta})^{W_n} = \sum_{\gamma} g^{\gamma}_{\alpha \beta} \chi^{\gamma}. \] By construction the character $\chi^{\gamma}$ associated to a double partition $\gamma = (\gamma^0, \gamma^1)$ of $n$ is induced from an irreducible character $\phi^{\gamma}$ of an inertia group $T_{\gamma}$, which is isomorphic to the direct product $W_{|\gamma^0|} \times W_{|\gamma^1|}$. More precisely \[ \phi^{\gamma} = \chi^{\gamma^0} \times \epsilon \chi^{\gamma^1}, \] where the characters $\chi^{\gamma^i}$, $i = 0, 1$, of the symmetric groups $\S_{|\gamma^i|}$ here are regarded as characters of $W_{|\gamma^i |}$ and where $\epsilon$ is the linear character of $W_{|\gamma^1|}$ which maps the sign change on $-1$ and the other generators on $1$. Similar statements hold for $\chi^{\alpha}$ and $\chi^{\beta}$. Since induction of characters commutes with the outer tensor product (s.~\cite{CR}, (43.2)) we have \[ \chi^{\alpha} \times \chi^{\beta} = (\phi^{\alpha})^{W_{n-k}} \times (\phi^{\beta})^{W_k} = (\phi^{\alpha} \times \phi^{\beta})^{W_{n-k} \times W_k}. \] Moreover \[ \phi^{\alpha} \times \phi^{\beta} = \chi^{\alpha^0} \times \epsilon \chi^{\alpha^1} \times \chi^{\beta^0} \times \epsilon \chi^{\beta^1} = (\chi^{\alpha^0} \times \chi^{\beta^0}) \times \epsilon (\chi^{\alpha^1} \times \chi^{\beta^1}), \] since $\epsilon \chi^{\alpha^1} \times \epsilon \chi^{\beta^1} = \epsilon (\chi^{\alpha^1} \times \chi^{\beta^1})$ for suitable characters $\epsilon$. Define $n_i = |\alpha_i| + |\beta_i|$ for $i = 0, 1$. Then \begin{multline*} \left((\chi^{\alpha^0} \times \chi^{\beta^0}) \times \epsilon (\chi^{\alpha^1} \times \chi^{\beta^1})\right)^{W_n} \\ = \left((\chi^{\alpha^0} \times \chi^{\beta^0})^{W_{n_0}} \times \epsilon (\chi^{\alpha^1} \times \chi^{\beta^1})^{W_{n_1}}\right)^{W_n}. \end{multline*} By the Littlewood-Richardson rule (\ref{thm:LR}) we have \[ (\chi^{\alpha^0} \times \chi^{\beta^0})^{W_{n_0}} = \sum_{\gamma^0} g^{\gamma^0}_{\alpha^0 \beta^0} \chi^{\gamma^0} \] and \[ \epsilon (\chi^{\alpha^1} \times \chi^{\beta^1})^{W_{n_1}} = \sum_{\gamma^1} g^{\gamma^1}_{\alpha^1 \beta^1} \epsilon \chi^{\gamma^1}, \] where in each case the sum is taken over all partitions $\gamma^i$ of $n_i$. So we finally have \begin{align*} (\chi^{\alpha} \times \chi^{\beta})^{W_n} & = (\phi^{\alpha} \times \phi^{\beta})^{W_n} \\ & = \Bigl(\sum_{\gamma^0} g^{\gamma^0}_{\alpha^0 \beta^0} \chi^{\gamma^0} \times \sum_{\gamma^1} g^{\gamma^1}_{\alpha^1 \beta^1} \epsilon \chi^{\gamma^1}\Bigr)^{W_n} \\ & = \sum_{\gamma^0, \gamma^1} g^{\gamma^0}_{\alpha^0 \beta^0} g^{\gamma^1}_{\alpha^1 \beta^1} (\chi^{\gamma^0} \times \epsilon \chi^{\gamma^1})^{W_n}, \end{align*} where by construction $(\chi^{\gamma^0} \times \epsilon \chi^{\gamma^1})^{W_n} = \chi^{\gamma}$. \end{proof} We now investigate the restriction of $\chi^{\gamma}$ to a subgroup of the form $W_{n-k} \times \S_k$. \begin{La} \label{la:LR-BA} Let $k \leq n$. For the restriction of the character $\chi^{\gamma}$ of $W_n$ on $W_{n-k} \times \S_k$ the following holds. \[ \chi^{\gamma}|_{W_{n-k} \times \S_k} = \sum_{\alpha, \beta, \delta} g^{\gamma}_{\alpha \beta} g^{\delta}_{\beta^0 \beta^1} \chi^{\alpha} \times \chi^{\delta}, \] where the sum is taken over all double partitions $\alpha$ of $n-k$, $\beta$ of $k$ and all partitions $\delta$ of $k$. \end{La} \begin{proof} By construction $\chi^{\beta}|_{\S_k} = (\chi^{\beta^0} \times \chi^{\beta^1})^{\S_k} = \sum_{\delta} g^{\delta}_{\beta^0 \beta^1} \chi^{\delta}$, where the sum is taken over all partitions $\delta$ of $k$ and the coefficients $g^{\delta}_{\beta^0 \beta^1}$ are given by the Littlewood-Richardson rule (\ref{thm:LR}). The assertion then follows from the above Proposition (\ref{thm:LR-B}). \end{proof} If the components of the skew diagram $\gamma \setminus \alpha$ are in a suitable way drawn one on top of the other the resulting diagram looks like a skew diagram of ordinary partitions. So it is possible to talk about tableaux of the form $\gamma \setminus \alpha$ with content $\beta$ for an ordinary partition $\beta$ of $n$. \begin{Prop} \label{thm:LR-BA} For double partitions $\alpha$ of $n-k$, $\gamma$ of $n$ and a partition $\beta = [k-r, 1^r]$ of $k$ the multiplicity $g^{\gamma}_{\alpha \beta}$ equals the number of skew standard tableaux of shape $\gamma \setminus \alpha$ and of content $\beta$, which yield lattice permutations if the entries are read row by row from right to left. In particular $g^{\gamma}_{\alpha\beta} = 0$ if $\alpha \not\subseteq \gamma$. \end{Prop} \begin{proof} By Lemma (\ref{la:LR-BA}) we have \[ g^{\gamma}_{\alpha \beta} = \sum_{\delta} g^{\gamma^0}_{\alpha^0 \delta^0} g^{\gamma^1}_{\alpha^1 \delta^1} g^{\beta}_{\delta^0 \delta^1}, \] where the sum is taken over all double partitions $\delta$ of $k$. We examine the nonzero summands. For this let $g^{\beta}_{\delta^0 \delta^1} > 0$. Then $\delta^i \subseteq \beta$ for $i = 0,1$, and since $\beta = [k-r, 1^r]$ is a hook, the $\delta^i$ also must be hooks. Therefore $\gamma^i \setminus \alpha^i$ is a strip for $i = 0,1$. Denote by $G^{\pi^1}_{\pi^2 \pi^3}$ the set of all standard tableaux of shape $\pi^1 \setminus \pi^2$, which are filled with content $\pi^3$ according to the Littlewood-Richardson rule. Under these conditions there is a bijection $\sigma$ between the set \[ \bigcup_{\delta} \left\{(T_0, T_1, T_2) \mid T_i \in G^{\gamma^i}_{\alpha^i \beta^i}, i = 0, 1, \text{ and } T_2 \in G^{\beta}_{\delta^0 \delta^1}\right\} \] and the set of all tableaux $(T'_0, T'_1)$ of shape $\gamma \setminus \alpha$ and of content $\beta$. For $\sigma(T_0, T_1, T_2) = (T'_0, T'_1)$ we have here $T'_0 = T_0$, and $T'_1$ arises from $T_1$ by replacing the entries in a suitable way, such that $(T'_0, T'_1)$ has content $\beta$. \end{proof} {\em Remark.} Presumably a suitable generalization of Proposition (\ref{thm:LR-B}) holds in arbitrary wreath products $C_r \wr \S_n$. Moreover Proposition (\ref{thm:LR-BA}) certainly is valid for arbitrary partitions $\beta$ and arbitrary $r$-tuples of partitions $\gamma$ and $\alpha$. For this article, however, it is sufficient to have both propositions in the proven form. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Representing Matrices.} \label{sec:matrix} Ariki and Koike (\cite{ArKo} (3.6)) give a method to construct matrices for all irreducible representations of $H^{(r)}_n$. This is a generalization of Hoefsmit's method (see~\cite{Hoefsmit} (2.2.6)) for the ordinary Iwahori-Hecke algebra of type $B_n$. For this let $\gamma$ be an $r$-tuple of partitions of $n$, furthermore let $f^{\gamma} = \chi^{\gamma}(1)$ be the degree of the corresponding character of $C_r \wr \S_n$ and let $\{\T_p \mid p = 1, \dots, f^{\gamma}\}$ be the set of all standard tableaux of shape $\gamma$. These tableaux form the basis of the $H^{(r)}_n$-module $V^{\gamma}$ defined below. For $k \in \Z$ and $y \in K$ let \[ \Delta(k, y) = 1 - q^k y \] and let \[ M(k, y) = \frac{1}{\Delta(k, y)} \left[\begin{array}{cc} q-1 & \Delta(k+1, y) \\ q \Delta(k-1, y) & -q^k y (q-1) \end{array} \right] \] be a $2 \times 2$-matrix over $K$. The action of $T$ on $V^{\gamma}$ is diagonal and given by \[ \T_p T = Q_{\tau(\T_p:1)} \T_p \quad \text{for all $p = 1, \dots, f^{\gamma}$,} \] where the entry $1$ lies in the component $\T^{\tau(\T_p:1)}_p$ of $\T_p$. The action of $S_i$ on $V^{\gamma}$ is given by \[ \T_p S_i = q \T_p, \] if $i$ and $i+1$ lie in the same row of a component of $\T_p$, and by \[ \T_p S_i = -\T_p, \] if $i$ and $i+1$ lie in the same column of a component of $\T_p$. Otherwise $\T_q:= \T_p (i, i+1)$ is again a standard tableau and $S_i$ acts on the subspace with basis $(\T_p, \T_q)$ as the matrix \[ M(r(i+1, i), Q_{\tau(\T_p:i)}/Q_{\tau(\T_p:i+1)}). \] Here $r(i+1, i)$ is the axial distance between $i+1$ and $i$ in $\T_p$, and $i$ (resp. $i+1$) lies in the component $\T^{\tau(\T_p:i)}_p$ (resp.~$\T^{\tau(\T_p:i+1)}_p$) of $\T_p$. The axial distance between $i+1$ and $i$ in $\T_q$ equals $-r(i+1, i)$, and \begin{multline*} M(-r(i+1, i), Q_{\tau(\T_p:i+1)}/Q_{\tau(\T_p:i)}) \\ = \left[{}^0_1 {\ }^1_0\right] M(r(i+1, i), Q_{\tau(\T_p:i)}/Q_{\tau(\T_p:i+1)}) \left[{}^0_1 {\ }^1_0\right] \end{multline*} So the action of $S_i$ on $V^{\gamma}$ is uniquely defined. \begin{Prop} The $V^{\gamma}$ form a complete system of absolutely irreducible, pairwise non-equivalent representations of $H^{(r)}_n$, if $\gamma$ runs through all $r$-tuples of partitions of $n$. \end{Prop} \begin{proof} \cite{ArKo} (3.7), (3.10). \end{proof} By the above definition $T_0 = T$ acts in each of the given representations as a diagonal matrix. It turns out that all $T_i$, $i = 0, \dots, n-1$, act in such a way. \begin{Prop} \label{thm:ti-op} Let $i \leq n-1$ and let $\T$ be a standard tableau of shape $\gamma$. Then \[ \T T_i = Q_{\tau(\T:i+1)} q^{c(\T:i+1)+i} \T, \] where $c(\T:i)$ denotes the content of $i$ in the tableau $\T$. \end{Prop} \begin{proof} \cite{ArKo} (3.16), cf. \cite{Hoefsmit} (3.3.3). \end{proof} For each $h \in H^{(r)}_n$ the matrix of the action of $h$ on $V^{\gamma}$ with respect to the basis of standard tableaux of shape $\gamma$ is denoted by $M^{\gamma}(h)$. The character of $h \in H^{(r)}_n$ on $V^{\gamma}$ is the trace of $M^{\gamma}(h)$ and is denoted by $\chi_q^{\gamma}(h)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Subalgebras.} \label{sec:sub} The following notation will be used throughout the rest of this article. Let $k \leq n$ and set \begin{align*} H_{n-k}^{(r)} & = \left, \\ H_k' & = \left, \\ H_k^{*} & = \left. \end{align*} So $H_{n-k}^{(r)}$ is a subalgebra of type $B_{n-k}^{(r)}$, $H_k' \cong H_k$ is a subalgebra of type $A_{k-1}$, and $H_k^{*}$ corresponds to a subalgebra of type $B_k$ (cf. section~\ref{sec:bem-B}). Let $\alpha$ be an $r$-tuple of partitions of $n-k$ and define $V^{\gamma \setminus \alpha}$ to be the $K$-vector space with the set of all standard tableaux of the form $\gamma \setminus \alpha$ as its basis. By Lemma (\ref{la:st-bij}) $V^{\gamma}$ has as a vector space the structure of a direct sum of tensor products, \[ V^{\gamma} = \bigoplus_{\alpha} V^{\alpha} \otimes V^{\gamma \setminus \alpha}. \] We write $\T = \T_1 \otimes \T_2$ for the decomposition of a standard tableau $\T$ of shape $\gamma$ into a standard tableau $\T_1$ of shape $\alpha$ and a standard tableau $\T_2$ of shape $\gamma \setminus \alpha$ according to Lemma (\ref{la:st-bij}). We may assume that the set of all standard tableaux of shape $\gamma$ is ordered as follows. For this purpose we assume a linear ordering on the set of all $r$-tuples of partitions of $n-k$ and for each $r$-tuple of partitions $\alpha$ of $n-k$ a linear ordering on the set of all standard tableaux of shape $\alpha$ and a linear ordering on the set of all standard tableaux of shape $\gamma \setminus \alpha$. Let $\T$ and $\T'$ be standard tableaux of shape $\gamma$ with $\T = \T_1 \otimes \T_2$ for a $\T_1$ of shape $\alpha$ and $\T' = \T'_1 \otimes \T'_2$ for a $\T'_1$ of shape $\alpha'$. Then define $\T < \T'$, if \begin{enumerate} \item $\alpha < \alpha'$, or \item $\alpha = \alpha'$ and $\T_1 < \T'_1$, or \item $\alpha = \alpha'$ and $\T_1 = \T'_1$ and $\T_2 < \T'_2$. \end{enumerate} Now certain matrices have the structure of block diagonal matrices, where each block is the Kronecker product of two smaller matrices. More precisely we have the two following lemmas. \begin{La} \label{la:h1} Let $h \in H^{(r)}_{n-k}$. Then \[ M^{\gamma}(h) = \bigoplus_{\alpha} M^{\alpha}(h) \otimes \Id, \] where $\alpha$ runs over all $r$-tuples of partitions of $n-k$ with $\alpha \subseteq \gamma$ and $\Id$ denotes the $f^{\gamma \setminus \alpha} \times f^{\gamma \setminus \alpha}$-identity matrix. \end{La} \begin{proof} For $\T = \T_1 \otimes \T_2$ we have by the definition of the action in section~\ref{sec:matrix} \[ \T T = Q_{\tau(\T:1)} (\T_1 \otimes \T_2) = \T_1 T \otimes \T_2, \] since $1$ lies in the same component of $\T_1$ as in $\T$, whence $\tau(\T:1) = \tau(\T_1:1)$. For all $i \leq n-k$ we have $\tau(\T:i) = \tau(\T_1:i)$ and $c(\T:i) = c(\T_1:i)$. Therefore the axial distance $r(i+1, i)$ is the same in $\T$ and $\T_1$. Moreover $\T (i, i+1) = \T_1 (i, i+1) \otimes \T_2$ for all $i < n-k$. Consequently \[ \T S_i = \T_1 S_i \otimes \T_2 \quad \text{for all $i < n-k$.} \] Thus $h \in H^{(r)}_{n-k}$ acts on each subspace $V^{\alpha} \otimes \T_2$ of $V^{\gamma}$ as the matrix $M^{\alpha}(h)$ according to section~\ref{sec:matrix}. Due to the ordering on the basis of $V^{\gamma}$ the matrix $M^{\gamma}(h)$ is a block diagonal matrix. \end{proof} \begin{La} \label{la:h2} Let $h \in H_k^{*}$. Then for each $r$-tuple of partitions $\alpha$ of $n-k$ there is a matrix $M^{\gamma \setminus \alpha}(h)$, such that \[ M^{\gamma}(h) = \bigoplus_{\alpha} \Id \otimes M^{\gamma \setminus \alpha}(h), \] where $\alpha$ runs over all $r$-tuple of partitions of $n-k$ with $\alpha \subseteq \gamma$ and $\Id$ denotes the $f^{\alpha} \times f^{\alpha}$-identity matrix. The map $h \mapsto M^{\gamma \setminus \alpha}(h)$ defines an action of $H_k^{*}$ on $V^{\gamma \setminus \alpha}$. \end{La} \begin{proof} Let $\T = \T_1 \otimes \T_2$ be a standard tableau of the form $\gamma$. By (\ref{thm:ti-op}) we have \begin{align*} (\T_1 \otimes \T_2) T_{n-k} & = Q_{\tau(\T:n-k+1)} q^{c(\T:n-k+1)+n-k} (\T_1 \otimes \T_2) \\ & = \T_1 \otimes Q_{\tau(\T_2:n-k+1)} q^{c(\T_2:n-k+1)+n-k} \T_2. \end{align*} For $i \geq n-k$ we have $\tau(\T:i) = \tau(\T_2:i)$ and $c(\T:i) = c(\T_2:i)$. Therefore the axial distance $r(i+1, i)$ between $i+1$ and $i$ is the same in $\T$ and in $\T_2$. Moreover $\T (i, i+1) = \T_1 \otimes \T_2 (i, i+1)$. Thus for each $r$-tuple of partitions $\alpha$ of $n-k$ and each $\T_1$ of shape $\alpha$ there is a matrix $M^{\gamma \setminus \alpha}(h)$, such that $h \in H_k^{*}$ acts on the subspace $\T_1 \otimes V^{\gamma \setminus \alpha}$ as $M^{\gamma \setminus \alpha}(h)$. Due to the ordering on the basis of $V^{\gamma}$ the matrix $M^{\gamma}(h)$ is a block diagonal matrix. Since $M^{\gamma}$ is a matrix representation of $H^{(r)}_n$ also $M^{\gamma \setminus \alpha}$ is a matrix representation of $H^{*}_k$. \end{proof} We denote by $\chi_q^{\gamma \setminus \alpha}$ the character of $H^{*}_k$ on $V^{\gamma \setminus \alpha}$. The above two lemmas can then be combined to the following crucial result on a decomposition of the character $\chi^{\gamma}_q$. \begin{Thm} \label{cor:tensor-sum} Let $h_1 \in H^{(r)}_{n-k}$, $h_2 \in H^{*}_k$ and let $\gamma$ be an $r$-tuple of partitions of $n$. Then \[ \chi^{\gamma}_q(h_1 h_2) = \sum_{|\gamma \setminus \alpha| = k} \chi^{\alpha}_q(h_1) \chi^{\gamma \setminus \alpha}_q(h_2), \] where the sum ranges over all $r$-tuples of partitions $\alpha$ of $n-k$ with $\alpha \subseteq \gamma$. \end{Thm} \begin{proof} The matrix $M^{\gamma}(h_1 h_2)$ is a block diagonal matrix since by (\ref{la:h1}) and (\ref{la:h2}) \begin{align*} M^{\gamma}(h_1 h_2) & = M^{\gamma}(h_1) M^{\gamma}(h_2) \\ & = \bigoplus_{\alpha} (M^{\alpha}(h_1) \otimes I)(I \otimes M^{\gamma \setminus \alpha}(h_2)) \\ & = \bigoplus_{\alpha} M^{\alpha}(h_1) \otimes M^{\gamma \setminus \alpha}(h_2), \end{align*} where each block belongs to an $r$-tuple of partitions $\alpha$ of $n-k$ and is a Kronecker product $M^{\alpha}(h_1) \otimes M^{\gamma \setminus \alpha}(h_2)$. Therefore the character value is \[ \chi_q^{\gamma}(h_1 h_2) = \Spur(M^{\gamma}(h_1 h_2)) = % \Spur(\bigoplus_{\alpha} (M^{\alpha}(h_1) \otimes M^{\gamma \setminus \alpha}(h_2))) = % \sum_{\alpha} \Spur(M^{\alpha}(h_1)) \Spur(M^{\gamma \setminus \alpha}(h_2)) = \sum_{\alpha} \chi^{\alpha}_q(h_1) \chi_q^{\gamma \setminus \alpha}(h_2). \] \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Restriction to Type $B \times A$.} \label{sec:res-BA} From now on let $r = 2$. Then $H^{(2)}_n$ is the Iwahori-Hecke algebra of type $B_n$. Remember that here $Q_1 = -1$ and $Q_0 = Q$. % Moreover Matsumoto's Lemma (\ref{la:matsu}) is valid for all elements $w \in W$. Let $k \leq n$, \[ \kappa = s_{n-k+1} \dots s_{n-1} \in W_n \quad \text{and so} \quad T_{\kappa} = S_{n-k+1} \dots S_{n-1} \in H. \] In this section we investigate the character values of elements of the form $h T_\kappa$ and in the next section the character values of elements of the form $h T_{n-k} T_\kappa$ for some $h \in H^{(2)}_{n-k}$. We recall the following result about character values on Coxeter elements in the Iwahori-Hecke algebra $H_n$ of type $A_{n-1}$. \begin{Prop} \label{thm:cox-A} Let $\gamma$ be a partition of $n$. Then \[ \chi^{\gamma}_q(T_{s_1} \dots T_{s_{n-1}}) = \begin{cases} (-1)^r q^{n-r-1} & \text{if $\gamma = [n-r, 1^r]$,} \\ 0 & \text{otherwise.} \end{cases} \] \end{Prop} \begin{proof} \cite{Pfeiffer94b} (2.2) \end{proof} \begin{La} \label{la:xga-BA} Let $\gamma$ be a double partition of and let $\alpha$ be a double partition of $n-k$. Then \[ \chi^{\gamma \setminus \alpha}_q(T_{\kappa}) = \begin{cases} (q-1)^{c^{\gamma}_{\alpha}-1} (-1)^{l_{\alpha}^{\gamma}} q^{k-l^{\gamma}_{\alpha}-c^{\gamma}_{\alpha}}, & \text{if $\gamma \setminus \alpha$ is a strip,} \\ 0 & \text{otherwise.} \end{cases} \] \end{La} \begin{proof} $\chi_q^{\gamma \setminus \alpha}$ is defined as a character of $H^*_k$ in section~\ref{sec:sub}. By restriction to $H'_k$ we obtain $\chi_q^{\gamma \setminus \alpha} = \sum_{\beta} g^{\gamma}_{\alpha\beta} \chi_q^{\beta}$, where the sum is over all partitions $\beta$ of $k$ and the coefficients $g^{\gamma}_{\alpha \beta}$ are given by the Littlewood-Richardson rule (\ref{thm:LR-BA}). We do not need, however, all partitions $\beta$ in order to determine the value of $\chi_q^{\gamma \setminus \alpha}(T_{\kappa})$. By (\ref{thm:cox-A}) we have $\chi_q^{\beta}(T_{\kappa}) = 0$, unless $\beta = [k-r, 1^r]$ for some $r \leq k-1$. In that case \[ \chi_q^{\beta}(T_{\kappa}) = (-1)^r q^{k-r-1}. \] Therefore \[ \chi_q^{\gamma \setminus \alpha}(T_{\kappa}) = \sum_{r=0}^{k-1} g^{\gamma}_{\alpha,[k-r,1^r]} (-1)^r q^{k-r-1}. \] For $\beta = [k-r, 1^r]$ the multiplicity $g^{\gamma}_{\alpha\beta}$ is by (\ref{thm:LR-BA}) equal to the number of skew semistandard tableaux of shape $\gamma \setminus \alpha$ and content $\beta$, which yield lattice permutations if the entries are read row by row from right to left. This means to place $k-r$ symbols $1$ and the symbols $2, \ldots r+1$ in a diagram of shape $\gamma \setminus \alpha$. This is impossible if the diagram $\gamma \setminus \alpha$ contains a $2 \times 2$-block. Hence $g_{\alpha, [k-r, 1^r]}^{\gamma} \not= 0$ only if $\gamma \setminus \alpha$ is a strip. So let $\gamma \setminus \alpha$ be a strip. Then we write \[ G(\gamma, \alpha):= \sum_{r=1}^{k-1} g_{\alpha, [k-r, 1^r]}^{\gamma} (-1)^r q^{k-r-1} = \sum_{\T} (-1)^{r(\T)} q^{k-r(\T)-1}, \] where the sum ranges over all admissible tableaux $\T$ according to the Littlewood-Richardson rule and $r(\T) = r$ if $\T$ has content $\beta = [k-r, 1^r]$. Since $\gamma \setminus \alpha$ is a strip, its connected components are hooks. The first of these has the following form. \begin{center} \setlength{\unitlength}{0.01in}% \begin{picture}(260,100)(20,700) \put(200,780){\framebox(60,20){}} \put(140,760){\framebox(80,20){}} \put( 80,740){\framebox(80,20){}} \put( 20,700){\framebox(80,20){}} \put( 65,725){\makebox(0.4444,0.6667){.}} \put( 70,730){\makebox(0.4444,0.6667){.}} \put( 75,735){\makebox(0.4444,0.6667){.}} \put( 40,720){\line( 0,-1){ 20}} \put( 60,720){\line( 0,-1){ 20}} \put( 80,720){\line( 0,-1){ 20}} \put(100,760){\line( 0,-1){ 20}} \put(120,760){\line( 0,-1){ 20}} \put(140,760){\line( 0,-1){ 20}} \put(160,780){\line( 0,-1){ 20}} \put(180,780){\line( 0,-1){ 20}} \put(200,780){\line( 0,-1){ 20}} \put(220,800){\line( 0,-1){ 20}} \put(240,800){\line( 0,-1){ 20}} \put(260,800){\line( 1, 0){ 20}} \put(280,800){\line( 0,-1){ 20}} \put(280,780){\line(-1, 0){ 20}} \put( 85,745){$1$} \put(125,745){$1$} \put(145,765){$1$} \put(185,765){$1$} \put(205,785){$1$} \put(245,785){$1$} \put(225,785){...} \put(165,765){...} \put(105,745){...} \put( 45,705){...} \put(205,765){$2$} \put(145,745){$3$} \put( 85,705){$s$} \put( 65,705){$1$} \put( 25,705){$1$} \put(265,785){$1$} \end{picture} \end{center} If $\gamma \setminus \alpha$ is connected, then there is only one admissible tableau $\T$ of that shape with $r(\T) = l^{\gamma}_{\alpha} = s-1$. Hence \[ G(\gamma, \alpha) = (-1)^{l^{\gamma}_{\alpha}} q^{k-l^{\gamma}_{\alpha}-1} \] which proves the assertion in that case. If $\gamma \setminus \alpha$ has more than one connected component, then each of the remaining connected components has one of the following two forms. \begin{center} \setlength{\unitlength}{0.01in}% \begin{picture}(270,220)(30,600) \put( 40,720){\framebox(80,20){}} \put(100,760){\framebox(80,20){}} \put(160,760){\line( 0, 1){ 40}} \put(160,800){\line( 1, 0){ 60}} \put(220,800){\line( 0, 1){ 20}} \put(220,820){\line( 1, 0){ 80}} \put(300,820){\line( 0,-1){ 20}} \put(300,800){\line(-1, 0){ 80}} \put(220,800){\line( 0,-1){ 20}} \put(220,780){\line( 1, 0){ 20}} \put(240,780){\line( 0, 1){ 40}} \put(180,800){\line( 0,-1){ 20}} \put(180,780){\line( 1, 0){ 40}} \put(260,820){\line( 0,-1){ 20}} \put(280,820){\line( 0,-1){ 20}} \put(200,800){\line( 0,-1){ 20}} \put(140,780){\line( 0,-1){ 20}} \put(120,780){\line( 0,-1){ 20}} \put(100,740){\line( 0,-1){ 20}} \put( 80,740){\line( 0,-1){ 20}} \put( 60,740){\line( 0,-1){ 20}} \put( 85,745){.} \put( 90,750){.} \put( 95,755){.} \put(225,805){$1$} \put(245,805){...} \put(265,805){$1$} \put(165,785){$1$} \put(185,785){...} \put(205,785){$1$} \put(105,765){$1$} \put(145,765){$1$} \put( 45,725){$1$} \put( 85,725){$1$} \put( 65,725){...} \put(125,765){...} \put( 40,600){\framebox(80,20){}} \put(100,640){\framebox(80,20){}} \put(160,640){\line( 0, 1){ 40}} \put(160,680){\line( 1, 0){ 60}} \put(220,680){\line( 0, 1){ 20}} \put(220,700){\line( 1, 0){ 80}} \put(300,700){\line( 0,-1){ 20}} \put(300,680){\line(-1, 0){ 80}} \put(220,680){\line( 0,-1){ 20}} \put(220,660){\line( 1, 0){ 20}} \put(240,660){\line( 0, 1){ 40}} \put(180,680){\line( 0,-1){ 20}} \put(180,660){\line( 1, 0){ 40}} \put(260,700){\line( 0,-1){ 20}} \put(280,700){\line( 0,-1){ 20}} \put(200,680){\line( 0,-1){ 20}} \put(140,660){\line( 0,-1){ 20}} \put(120,660){\line( 0,-1){ 20}} \put(100,620){\line( 0,-1){ 20}} \put( 80,620){\line( 0,-1){ 20}} \put( 60,620){\line( 0,-1){ 20}} \put( 85,625){.} \put( 90,630){.} \put( 95,635){.} \put(225,685){$1$} \put(245,685){...} \put(265,685){$1$} \put(165,665){$1$} \put(185,665){...} \put(205,665){$1$} \put(105,645){$1$} \put(145,645){$1$} \put( 45,605){$1$} \put( 85,605){$1$} \put( 65,605){...} \put(125,645){...} \put(285,805){$1$} \put(285,685){$i$} \put(221,665){$\scriptstyle i+1$} \put(161,645){$\scriptstyle i+2$} \put(101,605){$\scriptstyle s+1$} \put(105,725){$s$} \put( 10,770){(1)} \put( 10,650){(2)} \put(225,785){$i$} \put(161,765){$\scriptstyle i+1$} \end{picture} \end{center} Consider the diagram $\gamma \setminus \alpha'$ which consists of all but the last connected components of $\gamma \setminus \alpha$ for a suitable double partition $\alpha'$ and assume by induction on the number $c^{\gamma}_{\alpha}$ of connected components that \[ G(\alpha', \gamma) = \sum_{\T'} (-1)^{r(\T')} q^{k-r(\T')-1} = (q-1)^{c^{\gamma}_{\alpha'}-1} (-1)^{l^{\gamma}_{\alpha'}} q^{k'-c^{\gamma}_{\alpha'}-l^{\gamma}_{\alpha'}} \] where $k' = |\gamma \setminus \alpha'|$ and the sum ranges over all admissible tableaux $\T'$ for $\gamma \setminus \alpha'$ according to the Littlewood-Richardson rule. Note that $c^{\gamma}_{\alpha'} = c^{\gamma}_{\alpha}-1$. Observe that to each admissible tableau $\T'$ in the above summation there correspond exactly two admissible tableaux $\T_1$ and $\T_2$ for $\gamma \setminus \alpha$ with the following property. Both $\T_1$ and $\T_2$ consist of $\T'$ and an additional connected component which for $\T_1$ has form (1) and for $\T_2$ has form (2) and where $r(\T_2) = r(\T_1) + 1$. Their contribution to the sum $G(\gamma, \alpha)$ is \begin{multline*} (-1)^{r(\T_1)} q^{k-r(\T_1)-1} + (-1)^{r(\T_2)} q^{k-r(\T_2)-1} = (q-1) (-1)^{r(\T_1)} q^{k-r(\T_1)-2} \\ = (-1)^{r(\T')} q^{k'-r(\T')-1} \cdot (q-1) \cdot (-1)^{r(\T_1)-r(\T')} q^{k-k'-r(\T_1)+r(\T')-1}. \end{multline*} Finally $r(\T_1) - r(\T') = l_{\alpha}^{\gamma} - l_{\alpha'}^{\gamma}$ is independent of $\T'$ and therefore summation over all admissible $\T'$ for $\gamma \setminus \alpha'$ yields \begin{align*} G(\gamma, \alpha) & = G(\gamma, \alpha') (q-1) (-1)^{l^{\gamma}_{\alpha}-l^{\gamma}_{\alpha'}} q^{k-k'-l^{\gamma}_{\alpha}+l^{\gamma}_{\alpha'}-1} \\ & = (q-1)^{c^{\gamma}_{\alpha}-1} (-1)^{l_{\alpha}^{\gamma}} q^{k-l^{\gamma}_{\alpha}-c^{\gamma}_{\alpha}} \end{align*} which completes the proof of the lemma. \end{proof} The recursion formula for this case can now be stated as follows. \begin{Prop} \label{thm:form1} Let $h \in H_{n-k}^{(2)}$. Then \[ \chi_q^{\gamma}(h T_{\kappa}) = \sum_{|\gamma \setminus \alpha| = k} (q-1)^{c^{\gamma}_{\alpha}-1} (-1)^{l_{\alpha}^{\gamma}} q^{k-l^{\gamma}_{\alpha}-c^{\gamma}_{\alpha}} \chi_q^{\alpha} (h), \] where the sum is taken over all double partitions $\alpha \subseteq \gamma$ such that $\gamma \setminus \alpha$ is a strip. \end{Prop} \begin{proof} The assertion follows from the decomposition of the character $\chi_q^{\gamma} = \sum_{\alpha} \chi_q^{\alpha} \chi_q^{\gamma \setminus \alpha}$ in (\ref{cor:tensor-sum}) and the computation of $\chi_q^{\gamma \setminus \alpha}(T_{\kappa})$ in (\ref{la:xga-BA}). \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Restriction to Type $B \times B$.} \label{sec:res-BB} In contrast to the Weyl group there is in general no subalgebra of type $B_{n-k} \times B_k$ in the Iwahori-Hecke algebra of type $B_n$. Yet it is possible to pursue a similar strategy as in the preceding section. Let $\kappa$ and $T_{\kappa}$ be as in section~\ref{sec:res-BA} and let \[ W^{*}_k = \left \leq W_n. \] By specialization one obtains the following formula for wreath products from (\ref{cor:tensor-sum}). \begin{La} \label{la:spec-sum} Let $w_1 \in W_{n-k}$ and $w_2 \in W^{*}_{k}$. Then \[ \chi^{\gamma}(w_1 w_2) = \sum_{\alpha \subseteq \gamma} \chi^{\alpha}(w_1) \chi^{\gamma \setminus \alpha}(w_2). \] \end{La} \begin{Cor} \label{cor:schief} Let $\gamma$ be a double partition of $n$, let $\alpha$ be a double partition of $n-k$ and let $i < 2$. Then \[ \chi^{\gamma \setminus \alpha}(t^i_{n-k} \kappa) = \begin{cases} (-1)^{l^{\gamma}_{\alpha}} (-1)^{i j}, & \text{if $\gamma \setminus \alpha$ is a hook,} \\ 0 & \text{otherwise,} \end{cases} \] where $j = \tau(\gamma \setminus \alpha)$. \end{Cor} \begin{proof} This follows from the Murnaghan-Nakayama formula (\ref{cor:MNX}) for wreath products $C_r \wr \S_n$ of cyclic with symmetric groups and the regularity of the character table $(\chi^{\alpha}(w_1))$ of $W_{n-k}$ in Lemma (\ref{la:spec-sum}). \end{proof} We define \[ T_{\Delta} = \prod_{i=n-k}^{n-1} T_i \] and \[ e^{\gamma}_{\alpha} = \frac{c(\gamma) - c(\alpha)}{k} + n - \frac{k+1}{2} \] for double partitions $\alpha \subseteq \gamma$ with $|\gamma \setminus \alpha| = k$. Then $T_{\Delta}$ acts as a scalar on $V^{\gamma \setminus \alpha}$. \begin{La} \label{la:delta} Let $\T$ be a standard tableau of shape $\gamma \setminus \alpha$. Then \[ \T T_{\Delta} = q^{ke^{\gamma}_{\alpha}} \prod_{i=n-k}^{n-1} Q_{\tau(\T:i+1)} \T. \] \end{La} \begin{proof} By definition of $T_{\Delta}$ and Proposition (\ref{thm:ti-op}) we have \[ \T T_{\Delta} = \T \prod_{i=n-k}^{n-1} T_i = \prod_{i=n-k}^{n-1} \left(q^{c(\T:i+1)+i} Q_{\tau(i+1)}\right) \T. \] Here $\sum_{i=n-k}^{n-1} c(\T:i+1) = c(\gamma) - c(\alpha)$, which is the content of the skew diagram $\gamma \setminus \alpha$, and $\sum_{i=n-k}^{n-1} i = k n - k(k + 1)/2$, whence the sum of the exponents of $q$ equals $k e^{\gamma}_{\alpha}$. \end{proof} \begin{La} \label{la:root} $(T_{n-k} T_{\kappa})^k = T_{\Delta}$. \end{La} \begin{proof} For $k > 1$ and $i \geq n - k + 2$ we obtain (by the braid relations) \[ S_i (T_{n-k} T_{\kappa}) = (T_{n-k} T_{\kappa}) S_{i-1} \] and for $j < k$ \[ (T_{n-k} T_{\kappa})^j = (T_{n-k} S_{n-k+1} \dots S_{n-2})^j S_{n-1} \dots S_{n-j}. \] Therefore \begin{multline*} (T_{n-k} T_{\kappa})^k \\ = (T_{n-k} S_{n-k+1} \dots S_{n-2})^{k-1} S_{n-1} \dots S_{n-k+1} T_{n-k} S_{n-k+1} \dots S_{n-1}, \end{multline*} where the $(k-1)$-th power equals $T_{n-k} \dots T_{n-2}$ by induction and the rest equals $T_{n-1}$ by definition. \end{proof} In the case where $\gamma \setminus \alpha$ is a hook we define one further quantity $d^{\gamma}_{\alpha}$ as the content of a box immediately underneath the hook $\gamma \setminus \alpha$. Note that $e^{\gamma}_{\alpha}$ and $d^{\gamma}_{\alpha}$ depend on the actual choice of $\gamma$ and $\alpha$ and not only on the skew diagram $\gamma \setminus \alpha$. \begin{La} \label{la:chaken} If $\gamma \setminus \alpha$ is a hook, then $e^{\gamma}_{\alpha} = n + d^{\gamma}_{\alpha}$. \end{La} \begin{proof} A hook of length $k$ always occupies exactly one box in a row of $k$ successive diagonals. Hence \[ c(\gamma) - c(\alpha) = \sum_{i=1}^k (d^{\gamma}_{\alpha} + i) = k d^{\gamma}_{\alpha} + k (k + 1)/2, \] and the assertion follows from the definition of $e^{\gamma}_{\alpha}$. \end{proof} \begin{La} \label{la:xga-BB} We have \[ \chi_q^{\gamma \setminus \alpha}(T_{n-k} T_{\kappa}) = \begin{cases} Q_{\tau(\gamma\setminus\alpha)} (-1)^{l_{\alpha}^{\gamma}} q^{n+d^{\gamma}_{\alpha}}, & \text{if $\gamma \setminus \alpha$ is a hook,} \\ 0 & \text{otherwise.} \end{cases} \] \end{La} \begin{proof} Let \[ \rho = (\prod_{i=n-k}^{n-1} Q_{\tau(i+1)})^{1/k} e^{\gamma}_{\alpha}. \] By (\ref{la:root}) we have $T_{\Delta} = (T_{n-k} T_{\kappa})^k$ and $\rho^k$ is the only eigenvalue of $T_{\Delta}$ on $V^{\gamma \setminus \alpha}$ by (\ref{la:delta}). Therefore the eigenvalues of $T_{n-k} T_{\kappa}$ are of the form $\xi^j \rho$ for a $k$-th root of unity $\xi$ and certain $j = 0, \dots, k-1$. Hence $\chi_q^{\gamma \setminus \alpha}(T_{n-k} T_{\kappa}) = Z \rho$ for some sum $Z$ of $k$-th roots of unity. $\rho$ specializes to $(\prod (-1)^{\tau(i+1)})^{1/k}$, and by (\ref{cor:schief}) we have \begin{align*} Z \Bigl(\prod_{i=n-k}^{n-1} (-1)^{\tau(i+1)}\Bigr)^{1/k} &= \chi^{\gamma \setminus \alpha}(t_{n-k} \kappa) \\ &= \begin{cases} (-1)^{l^{\gamma}_{\alpha}} (-1)^{\tau(\gamma \setminus \alpha)} & \text{for hooks $\gamma \setminus \alpha$,} \\ 0 & \text{otherwise.} \end{cases} \end{align*} Since $\rho \not= 0$ we have $Z = 0$ and therefore $\chi_q^{\gamma \setminus \alpha}(T_{n-k} T_{\kappa}) = 0$, if $\gamma \setminus \alpha$ is not a hook. Otherwise all boxes of the diagram $\gamma \setminus \alpha$ lie in the same component $\tau(\gamma \setminus \alpha)$ of the diagram $\gamma$ and \[ \prod_{i=n-k}^{n-1} Q_{\tau(i+1)} = Q_{\tau(\gamma \setminus \alpha)}^k. \] Then $\rho = q^{e^{\gamma}_{\alpha}} Q_{\tau(\gamma \setminus \alpha)}$ and $\rho$ specializes to $(-1)^{\tau(\gamma \setminus \alpha)}$. Therefore $Z = (-1)^{l^{\gamma}_{\alpha}}$, and by (\ref{la:chaken}) we have $e^{\gamma}_{\alpha} = n + d^{\gamma}_{\alpha}$. \end{proof} The recursion formula in this case can now be stated as follows. \begin{Prop} \label{thm:form2} Let $h \in H_{n-k}^{(2)}$. Then \[ \chi_q^{\gamma}(h T_{n-k} T_{\kappa}) = \sum_{|\gamma \setminus \alpha| = k} Q_{\tau(\gamma\setminus\alpha)} (-1)^{l_{\alpha}^{\gamma}} q^{n+d^{\gamma}_{\alpha}} \chi_q^{\alpha}(h), \] where the sum is taken over all double partitions $\alpha$ of $n-k$ , such that $\gamma \setminus \alpha$ is a hook. Here $d^{\gamma}_{\alpha}$ denotes the content of a box directly underneath the hook $\gamma \setminus \alpha$. \end{Prop} \begin{proof} The assertion follows from the decomposition of the character $\chi_q^{\gamma} = \sum_{\alpha} \chi_q^{\alpha} \chi_q^{\gamma \setminus \alpha}$ in (\ref{cor:tensor-sum}) and the computation of $\chi_q^{\gamma \setminus \alpha}(T_{n-k} T_{\kappa})$ in (\ref{la:xga-BB}). \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The Character Table of the Iwahori-Hecke Algebra of Type $B_n$.} \label{sec:chq-B} We combine the formulas of the preceding two sections into a theorem which completely describes the character table of the Iwahori-Hecke algebra of type $B_n$. For this let $\chi_q^{([\ ], [\ ])} = 1$. \begin{Thm} \label{thm:chq-B} Let $\gamma = (\gamma^0, \gamma^1)$ and $\pi = (\pi^0, \pi^1)$ be double partitions of $n$ and let $w \in W_n$ be an element of minimal length in the class of element with cycle structure $\pi$. Furthermore let $\epsilon = 1$, if $\pi^0 =[\ ]$, and $\epsilon = 0$ otherwise. In the case $\epsilon = 0$ let $k = \pi^0_l$ for $l = l(\pi^0)$, otherwise let $k = \pi^1_1$. Finally let $\rho$ be the double partition of $n-k$, which results from $\pi$ by removing the part $k$. Then the value $\chi_q^{\gamma}(\pi)$ of the character $\chi_q^{\gamma}$ on $T_w$ is given by \[ \chi_q^{\gamma}(\pi) = \begin{cases} \displaystyle \sum_{|\gamma \setminus \alpha| = k} (q-1)^{c^{\gamma}_{\alpha}-1} (-1)^{l_{\alpha}^{\gamma}} q^{k-l^{\gamma}_{\alpha}-c^{\gamma}_{\alpha}} \chi_q^{\alpha} (\rho), & \text{if $\epsilon = 0$, and} \\ \displaystyle \sum_{|\gamma \setminus \alpha| = k} Q_{\tau(\gamma\setminus\alpha)} (-1)^{l_{\alpha}^{\gamma}} q^{n+d^{\gamma}_{\alpha}} \chi_q^{\alpha}(\rho) & \text{otherwise.} \end{cases} \] Here the sum is taken over all double partitions $\alpha$ of $n-k$, such that $\gamma \setminus \alpha$ is a strip in the case $\epsilon = 0$ and a hook in the case $\epsilon = 1$. \end{Thm} \begin{proof} The standard representative $w_{\pi}$ of the class with label $\pi$ which is constructed in section~\ref{sec:ptupel} consists of blocks of the form $t_i^{\epsilon} s_{i+1} \dots s_{i+j-1}$. Such a block corresponds for $\epsilon = 1$ to a part $j$ of $\pi^1$ and for $\epsilon = 0$ to a part $j$ of $\pi^0$. In~\cite{GePf93} it is shown, that these elements are of minimal length in their conjugacy class. The corresponding element $T_{w_{\pi}} \in H^{(2)}_n$ therefore has the form \[ T_{w_{\pi}} = T_{w_{\rho}} T_{n-k}^{\epsilon} S_{n-k+1} \dots S_{n-1} \] for some $T_{w_{\rho}} \in H^{(2)}_{n-k}$ whose character values are known by induction. In the case $\epsilon = 1$ we have $\pi^0 = [\ ]$ and $k = \pi^1_1$. Since $S_{n-k+1} \dots S_{n-1} = T_{\kappa}$ the assertion follows here by (\ref{thm:form2}) and $h = T_{w_{\rho}} \in H^{(2)}_{n-k}$. Otherwise $k = \pi^0_{l(\pi^0)}$ and the assertion follows by (\ref{thm:form1}) and $h = T_{w_{\rho}}$. \end{proof} The result is illustrated by character table of the Iwahori-Hecke algebra of type $B_2$ in Table~\ref{tab:ctq-b2} and that of the Iwahori-Hecke algebra of type $B_3$ in Table~\ref{tab:ctq-b3}. \begin{table}[htbp] \begin{center} \leavevmode \begin{tabular}{|r||c|c|c|c|c|} \hline $H(B_2)$ & $(1^2, 0)$ & $(1, 1)$ & $(0, 1^2)$ & $(2, 0)$ & $(0, 2)$ \\ & $1$ & $t$ & $t s t s$ & $s$ & $t s$ \\ \hline\hline $(1^2, 0)$ & $1$ & $Q$ & $Q^{2}$ & $-1$ & $-Q$ \\ \hline $(1, 1)$ & $2$ & $Q-1$ & $-2\,qQ$ & $q-1$ & $0$ \\ \hline $(0, 1^2)$ & $1$ & $-1$ & $1$ & $-1$ & $1$\\ \hline $(2, 0)$ & $1$ & $Q$ & $q^{2}Q^{2}$ & $q$ & $qQ$\\ \hline $(0, 2)$ & $1$ & $-1$ & $q^{2}$ & $q$ & $-q$ \\ \hline \end{tabular} \end{center} \caption{\strut The Iwahori-Hecke algebra of type $B_2$.} \label{tab:ctq-b2} \end{table} \begin{table}[hbtp] \begin{center} \leavevmode \begin{tabular}{|r||c|c|c|c|c|} \hline $H(B_3)$ & $(1^3,0)$ & $(1^2,1)$ & $(1,1^2)$ & $(0,1^3)$ & $(21,0)$ \\ & $1$ & $t_0$ & $t_0 t_1$ & $t_0 t_1 t_2$ & $s_1$ \\ \hline \hline $(1^3,0)$ & $1$ & $Q$ & $Q^2$ & $Q^3$ & $-1$ \\ \hline $(1^2,1)$ & $3$ & $2Q-1$ & $Q^2-2qQ$ & $-3q^2Q^2$ & $q-2$ \\ \hline $(1,1^2)$ & $3$ & $Q-2$ & $-2qQ+1$ & $3q^2Q$ & $q-2$ \\ \hline $(0,1^3)$ & $1$ & $-1$ & $1$ & $-1$ & $-1$ \\ \hline $(21,0)$ & $2$ & $2Q$ & $q^2Q^2+Q^2$ & $2q^3Q^3$ & $q-1$ \\ \hline $(1,2)$ & $3$ & $Q-2$ & $-2qQ+q^2$ & $3q^4Q$ & $2q-1$ \\ \hline $(2,1)$ & $3$ & $2Q-1$ & $q^2Q^2-2qQ$ & $-3q^4Q^2$ & $2q-1$ \\ \hline $(0,21)$ & $2$ & $-2$ & $q^2+1$ & $-2q^3$ & $q-1$ \\ \hline $(3,0)$ & $1$ & $Q$ & $q^2Q^2$ & $q^6Q^3$ & $q$ \\ \hline $(0,3)$ & $1$ & $-1$ & $q^2$ & $-q^6$ & $q$ \\ \hline \end{tabular} \vspace*{.3cm} \begin{tabular}{|r||c|c|c|c|c|} \hline $H(B_3)$ & $(1,2)$ & $(2,1)$ & $(0,21)$ & $(3,0)$ & $(0,3)$ \\ (cont.) & $t_0 s_1$ & $t_0 s_2$ & $t_0 t_1 s_2$ & $s_1 s_2$ & $t_0 s_1 s_2$ \\ \hline \hline $(1^3,0)$ & $-Q$ & $-Q$ & $-Q^2$ & $1$ & $Q$ \\ \hline $(1^2,1)$ & $-Q$ & $qQ-Q+1$ & $qQ$ & $-q+1$ & $0$ \\ \hline $(1,1^2)$ & $1$ & $-Q-q+1$ & $qQ$ & $-q+1$ & $0$ \\ \hline $(0,1^3)$ & $1$ & $1$ & $-1$ & $1$ & $-1$ \\ \hline $(21,0)$ & $qQ-Q$& $qQ-Q$ & $0$ & $-q$ & $-qQ$ \\ \hline $(1,2)$ & $-q$ & $qQ-q+1$ & $-q^2Q$ & $q^2-q$ & $0$ \\ \hline $(2,1)$ & $qQ$ & $qQ-Q-q$ & $-q^2Q$ & $q^2-q$ & $0$ \\ \hline $(0,21)$ & $-q+1$ & $-q+1$ & $0$ & $-q$ & $q$ \\ \hline $(3,0)$ & $qQ$ & $qQ$ & $q^3Q^2$ & $q^2$ & $q^2Q$ \\ \hline $(0,3)$ & $-q$ & $-q$ & $q^3$ & $q^2$ & $-q^2$ \\ \hline \end{tabular} \caption{\strut The Iwahori-Hecke algebra of type $B_3$.} \label{tab:ctq-b3} \end{center} \end{table} The rows in these tables correspond to the irreducible characters and the columns correspond to the conjugacy classes of the Weyl group. For each class a representative of minimal length is given. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Concluding Remarks.} \label{sec:bem-B} Proposition (\ref{thm:ti-op}) has an interesting consequence. \begin{Cor} \label{cor:rel} For $T_{n-1} = S_{n-1} \dots S_1 T S_1 \dots S_{n-1}$ we have \[ \prod_{i=0}^{2n-2} \prod_{j=0}^{r-1} (T_{n-1} - q^i Q_j) = 0. \] \end{Cor} \begin{proof} $T_{n-1}$ acts diagonal in every irreducible representation of $H^{(r)}_n$ and its eigenvalues are of the form $q^{c(\T:n)+n-1}Q_j$ by (\ref{thm:ti-op}). The content $c(\T:n)$ of $n$ in a standard tableau $\T$ of shape $\gamma$ for an $r$-tuple of partitions $\gamma$ of $n$ lies between $-(n-1)$ and $n-1$. Therefore $0 \leq c(\T:n) + (n-1) \leq 2n-2$. (For $r = 1$ and $n = 1,2,3$ further restrictions apply here.) \end{proof} \begin{Cor} $H^{*}_k$ is an epimorphic image of a specialization of the cyclotomic algebra of type $B^{(r(2(n-k)+1))}_k$. \end{Cor} \begin{proof} The generators $S_i$, $i = n-k+1, \dots, n-1$ of $H^{*}_k$ satisfy the defining relations (ii), (v) and (vi) of the cyclotomic algebra of type $B^{(r(2(n-k)+1))}_k$ in Definition (\ref{def:hck-B}). The generator $T_{n-k}$ of $H^{*}_k$ commutes with all $S_i$, $i > n-k+1$, (iv), and \begin{multline*} T_{n-k} S_{n-k+1} T_{n-k} S_{n-k+1} = T_{n-k} T_{n-k+1} = T_{n-k+1} T_{n-k} \\ = S_{n-k+1} T_{n-k} S_{n-k+1} T_{n-k}. \end{multline*} By (\ref{cor:rel}) $T_{n-k}$ also satisfies the relation \[ \prod_{i=0}^{2(n-k)} \prod_{j=0}^{r-1} (T_{n-k} - q^i Q_j) = 0. \] Therefore all defining relations of a cyclotomic algebra of type $B^{(r(2(n-k)+1))}_k$ with parameters $q$ and $q^i Q_j$, $i = 0, \dots, 2(n-k)$, $j = 0, \dots, r-1$ are satisfied. \end{proof} Moreover $H^{(r)}_{n-k}$ commutes with $H^{*}_k$. \begin{La} \label{la:decom} Let $h_1 \in H^{(r)}_{n-k}$ and $h_2 \in H^{*}_k$. Then $h_1 h_2 = h_2 h_1$. \end{La} \begin{proof} Because of the relations in Definition (\ref{def:hck-B}) it is sufficient to show that $T_{n-k}$ commutes with the generators of $H^{(r)}_{n-k}$. Let $1 \leq j < n-k$. Then we have (by the braid relations) \begin{align*} S_j T_{n-k} & = S_{n-k} \dots S_1 S_{j+1} T S_1 \dots S_{n-k} \\ & = S_{n-k} \dots S_1 T S_{j+1} S_1 \dots S_{n-k} = T_{n-k} S_j. \end{align*} Moreover \begin{align*} T \,T_{n-k} & = S_{n-k} \dots S_2 T S_1 T S_1 \dots S_{n-k} \\ & = S_{n-k} \dots S_1 T S_1 T S_2 \dots S_{n-k} = T_{n-k} T. \end{align*} \end{proof} This explains to what extent it is appropriate to talk in section~\ref{sec:res-BB} about a restriction to a subalgebra of type $B \times B$. For $r = 1$ (and $Q_0 = 1$) the algebra $H^{(r)}_n$ is the Iwahori-Hecke-Algebra $H_n$ of type $A_{n-1}$. In this case the proof of (\ref{thm:form1}) together with the ordinary Littlewood-Richardson rule (\ref{thm:LR}) yields the formula for the character table of $H_n$ from~\cite{Pfeiffer94b}. \begin{Thm} \label{thm:mna} Let $\pi$ and $\gamma$ be partitions of $n$ and let $w \in \S_n$ be an element of minimal length in the class of elements with cycle structure $\pi$. Moreover let $k = \pi_i$ for some $i \leq l(\pi)$ and let $\rho = [\pi_1, \dots, \pi_{i-1}, \pi_{i+1}, \dots, \pi_{l(\pi)}]$. Then the value $\chi_q^{\gamma}(\pi)$ of the character $\chi_q^{\gamma}$ on $T_w$ is given by \[ \chi_q^{\gamma}(\pi) = \sum_{|\gamma \setminus \alpha| = k} (q-1)^{c^{\gamma}_{\alpha}-1} (-1)^{l_{\alpha}^{\gamma}} q^{k-l^{\gamma}_{\alpha}-c^{\gamma}_{\alpha}} \chi_q^{\alpha} (\rho), \] where the sum is over all partitions $\alpha$ of $n-k$ such that $\gamma \setminus \alpha$ is a strip. \end{Thm} The proof of (\ref{thm:form2}) also works for $r > 2$. After a further generalization of the Littlewood-Richardson rules in section~\ref{sec:LR-B} for wreath products $C_r \wr \S_n$ the proof of (\ref{thm:form1}) also works for elements of the form $h T_{\kappa}$ in a cyclotomic algebra $H^{(r)}_n$ with $r > 2$. Then both character formulas would be valid in general in cyclotomic algebras $H^{(r)}_n$. 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