The Iwahori-Hecke algebra of type $B_n$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 10 June 2010

The Iwahori-Hecke algebra of type $B_n$

Fix $p,q\in {ℂ}^{*}.$ The Iwahori-Hecke algebra $H$ of type $B_n$ is the algebra given by generators $T_1,\dots ,T_n$ and relations $$T_i{T}_j=T_i{T}_j,\text{if }\left|i-j\right|>1,$$$$T_1{T}_2{T}_1{T}_2=T_2{T}_1{T}_2{T}_1\text{ and }{T}_i{T}_{i+1}{T}_i=T_{i+1}{T}_i{T}_{i+1}$$$$T_1^2=\left(p-p^{-1}\right)T_1+1\text{ and }{T}_i^2=\left(q-q^{-1}\right)T_i+1,2\le i\le n.$$

The Iwahori-Hecke algebra $H{B}_n$ has dimension $\le 2^{n}n!$.

1. The irreducible representations $H^{\left(\alpha ,\beta \right)}$ of $H{B}_n$ are indexed by pairs $\left(\alpha ,\beta \right)$ with $n$ boxes in total.
2. $\dim H^{\left(\alpha ,\beta \right)}=$# of standard tableaux of shape $\left(\alpha ,\beta \right).$
3. The irreducible $H{B}_n$-module $H^{\left(\alpha ,\beta \right)}$ is given by $$S^{\left(\alpha ,\beta \right)}=ℂ-\mathrm{span}\left\{v_T|T\text{ is a standard tableau of shape }\left(\alpha ,\beta \right)\right\}$$ with basis $\left\{v_T\right\}$ and with $H{B}_n$ action given by $$T_1{v}_T=\mathrm{sgn}\left(T\left(1\right)\right)p^{\mathrm{sgn}\left(T\left(1\right)\right)}{q}^{c\left(T\left(1\right)\right)}{v}_T,$$$$T_i{v}_T={\left(T_i\right)}_{TT}{v}_T+\left(1+{\left(T_i\right)}_{TT}\right)v_{s_{i}T},i=2,3,\dots ,n,$$ where $${\left(T_i\right)}_{TT}=\frac{q-q^{-1}}{1-\left(CT\left(T\left(k-1\right)\right)\right)/CT\left(T\left(k\right)\right)},\text{with }CT\left(T\left(k\right)\right)=\mathrm{sgn}\left(T\left(k\right)\right)p^{\mathrm{sgn}\left(T\left(k\right)\right)}{q}^{c\left(T\left(k\right)\right)},$$
   1. $clt\left(T\left(i\right)\right)$ is the content of the box containing $i$ in $T,$
   2. $\mathrm{sgn}\left(T\left(i\right)\right)$ is the sign of the box containing $i$ in $T,$
   3. $s_{i}T$ is the same as $T$ except that $i$ and $i-1$ are switched, and
   4. $v_{s_{i}T}=0,$ if $s_{i}T$ is not a standard tableau.

		Proof.
	Let us show that the action satisifes the given relations. The relations $T_i{T}_j=T_j{T}_i,$$\left|i-j\right|>1,ij\ge 2,$ and $T_i{T}_{i+1}{T}_i=T_{i+1}{T}_i{T}_{i+1},2\le i\le n,$ are taken care of by the case of the Iwahori-Hecke algebra of type $A.\square$

Define $$X^{{\epsilon }_1}=T_1,V^{{\epsilon }_1}=T_i{T}_{i-1}\dots T_2{T}_1{T}_2\dots T_{i-1}{T}_i,$$ in $H{B}_n.$ Then the action of $X^{{\epsilon }_1},1\le i\le n,$ on the $H{B}_n$-module $H^{\left(\alpha ,\beta \right)}$ is given by $$X^{{\epsilon }_1}{v}_T=CT\left(T\left(i\right)\right)v_T.$$

		Proof.
	By the definition of the action $T_1{V}_t=CT\left(T\left(1\right)\right)v_T=X^{{\epsilon }_1}{v}_T$ and, by induction, $$\begin{array}{rcl}{X}^{{\epsilon }_1}& =& T_i{X}^{{\epsilon }_{i-1}}{T}_i{v}_T\\ & =& T_i\left(CT\left(T\left(i-1\right)\right){\left(T_k\right)}_{TT}{v}_T+CT\left(T\left(i\right)\right)\left(q^{-1}+{\left(T_k\right)}_{TT}{v}_{s_{k}T}\right)\right)\\ & =& CT\left(T\left(i\right)\right)T_i\left(T_i-\left(q-q^{-1}\right)\right)v_T\\ & =& CT\left(T\left(i\right)\right)v_T.\square \end{array}$$

Note that $$\begin{array}{rcl}1-\frac{CT\left(T\left(i-1\right)\right)}{CT\left(T\left(i\right)\right)}& =& 1-\frac{\mathrm{sgn}\left(T\left(i-1\right)\right)}{\mathrm{sgn}\left(T\left(i\right)\right)}\frac{p^{\mathrm{sgn}\left(T\left(i-1\right)\right)}{q}^{c\left(T\left(i-1\right)\right)}}{p^{\mathrm{sgn}\left(T\left(i\right)\right)}{q}^{c\left(T\left(i\right)\right)}}\\ & =& \mathrm{sgn}\left(T\left(i\right)\right)p^{\mathrm{sgn}\left(T\left(i\right)\right)}{q}^{c\left(T\left(i\right)\right)}-\mathrm{sgn}\left(T\left(i-1\right)\right)p^{\mathrm{sgn}\left(T\left(i-1\right)\right)}{q}^{c\left(T\left(i-1\right)\right)}\end{array}$$ When $p=q,$ $$\begin{array}{rcl}\frac{q-q^{-1}}{1-\frac{CT\left(T\left(i-1\right)\right)}{CT\left(T\left(i\right)\right)}}& =& \frac{q-q^{-1}}{\mathrm{sgn}\left(T\left(i\right)\right)q^{\mathrm{sgn}\left(T\left(i\right)\right)}{q}^{c\left(T\left(i\right)\right)}-\mathrm{sgn}\left(T\left(i-1\right)\right)q^{\mathrm{sgn}\left(T\left(i-1\right)\right)}{q}^{c\left(T\left(i-1\right)\right)}}\\ & =& \frac{-\mathrm{sgn}\left(T\left(i-1\right)\right)q^{-c\left(T\left(i-1\right)\right)-\mathrm{sgn}\left(T\left(i-1\right)\right)}\left(q-q^{-1}\right)}{1-\mathrm{sgn}\left(T\left(i-1\right)\mathrm{sgn}\left(T\left(i\right)\right)q^{\mathrm{sgn}\left(T\left(i\right)\right)+c\left(T\left(i\right)\right)-\mathrm{sgn}\left(T\left(i-1\right)\right)-c\left(T\left(i-1\right)\right)}\right)}\\ & =& \left\{\begin{array}{rcl}1/\left[c\left(T\left(i\right)\right)-c\left(T\left(i-1\right)\right)\right],& \text{if}& \mathrm{sgn}\left(T\left(i\right)\right)=\mathrm{sgn}\left(T\left(i-1\right)\right),\\ \left(q-q^{-1}\right)/\left(1+q^{c\left(T\left(i\right)\right)-c\left(T\left(i-1\right)\right)\pm 2}\right),& \text{if}& \mathrm{sgn}\left(T\left(i\right)\right)=-\mathrm{sgn}\left(T\left(i-1\right)\right),\end{array}\right.\end{array}$$ and at $q=1$ this is $$\left\{\begin{array}{rcl}1/\left[c\left(T\left(i\right)\right)-c\left(T\left(i-1\right)\right)\right],& \text{if}& \mathrm{sgn}\left(T\left(i\right)\right)=\mathrm{sgn}\left(T\left(i-1\right)\right),\\ 0,& \text{if}& \mathrm{sgn}\left(T\left(i\right)\right)=-\mathrm{sgn}\left(T\left(i-1\right)\right),\end{array}\right.$$

Let $u_0,u_1,\dots ,u_{r-1}\in ℂ$ and let $q\in {ℂ}^{*}.$ The cyclotomic algebra $H_{r,1,n}$ is the algebra given by generators $X^{{\epsilon }_1},T_2,T_3,\dots ,T_n$ and relations $$T_i{T}_j=T_j{T}_i,\text{if }\left|i-j\right|>1,$$$$T_i{T}_{i+1}{T}_i=T_{i+1}{T}_i{T}_{i+1},2\le i\le n-1,$$$$X^{{\epsilon }_1}{T}_j=T_j{X}^{{\epsilon }_1},j>1,$$$$X^{{\epsilon }_1}{T}_2{X}^{{\epsilon }_1}{T}_2=T_2{X}^{{\epsilon }_1}{T}_2{X}^{{\epsilon }_1},$$$$\left(X^{{\epsilon }_1}-u_0\right)\left(X^{{\epsilon }_1}-u_1\right)\dots \left(X^{{\epsilon }_1}-u_{r-1}\right)=0,$$$$T_i^2=\left(q-q^{-1}\right)T_i+1,2\le i\le n.$$

If $p_j={\xi }^j$ where $\xi =e^{2\pi i/r}$ and $q=1$ then $H_{r,1,n}=ℂG\left(r,1,n\right).$ Define $$X^{{\epsilon }_k}=T_i{T}_{i-1}\dots T_2{X}^{{\epsilon }_1}{T}_2\dots T_{i-1}{T}_i,\text{for }1i\le n,$$ and $$X^{\lambda }={\left(X_1^{\epsilon }\right)}^{{\lambda }_1}\dots {\left(X_n^{\epsilon }\right)}^{{\lambda }_n},\text{ for }\lambda ={\lambda }_1{\epsilon }_1+\dots +{\lambda }_n{\epsilon }_n,{\lambda }_i\in \mathbb{Z}.$$ Then $$ℂ\left[X\right]=ℂ-\mathrm{span}\left\{X^{\lambda }|\lambda ={\lambda }_1{\epsilon }_1+\dots +{\lambda }_n{\epsilon }_n,{\lambda }_i\in \mathbb{Z}\right\}$$ is a commutative subalgebra of $H_{r,1,n}.$

If $w\in S_n,$ define $T_w=T_{i_1}\dots T_{i_p}$ if $w=s_{i_1}\dots s_{i_p}$ and $p$ is as small as possible. The cyclotomic Hecke algebra $H_{r,1,n}$ has basis $$X^{\lambda }{T}_w,\text{where }w\in S_n,\lambda ={\lambda }_1{\epsilon }_1+\dots +{\lambda }_n{\epsilon }_n,0\le {\lambda }_j\le r-1,$$ and $\dim \left(H_{r,1,n}\right)=r^{n}n!.$

1. The irreducible representations $H^{\lambda }$ of $H_{r,1,n}$ are indexed by $r$-tuples $\lambda =\left({\lambda }^{\left(0\right)},\dots ,{\lambda }^{\left(r-1\right)}\right)$ of partitions with $n$ boxes in total.
2. $\dim H^{\lambda }=$# of standard tableaux of shape $\lambda .$
3. The irreducible $H_{r,1,n}$-module $H^{\lambda }$ is given by $$H^{\lambda }=ℂ-\mathrm{span}\left\{v_T|T\text{ is a standard tableau of shape }\lambda =\left({\lambda }^{\left(0\right)},\dots ,{\lambda }^{\left(r-1\right)}\right)\right\}$$ with basis $\left\{v_T\right\}$ and with $H_{r,1,n}$ action given by $$\begin{array}{rcl}{X}^{{\epsilon }_1}{v}_T& =& CT\left(T\left(1\right)\right)v_T,\end{array}$$$$\begin{array}{rcl}{T}_i{v}_T& =& {\left(T,i\right)}_{TT}{v}_T+\left(q^{-1}+{\left(T_i\right)}_{TT}\right)v_{s_{i}T},i=2,3,\dots ,n,\end{array}$$ where $${\left(T_i\right)}_{TT}=\frac{q-q^{-1}}{1-\left(CT\left(T\left(k-1\right)\right)\right)/CT\left(T\left(k\right)\right)},\text{with }CT\left(b\right)=u_{s\left(b\right)}{q}^{2c\left(b\right)},$$
   1. $c\left(b\right)$ is the content of the box $b,$
   2. $T\left(i\right)$ is the box containing $i$ in $T,$
   3. $s_{i}T$ is the same as $T$ except that $i$ and $i-1$ are switched, and
   4. $v_{s_{i}T}=0,$ if $s_{i}T$ is not a standard tableau.

The action of the elements $X^{{\epsilon }_i},1\le i\le n,$ on $H^{\lambda }$ is given by $$X^{{\epsilon }_i}{v}_T=CT\left(T\left(i\right)\right)v_T,$$ for all standard tableaux $T$ of shape $\lambda .$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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