Standard Young tableaux, representations and Jucys-Murphy elements

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

Last update: 27 February 2013

Standard Young tableaux, representations and Jucys-Murphy elements

In this section we review the generalization of standard Young tableaux in [Ram0401326] which is used to construct representations of the affine Hecke algebras $H_{\infty ,1,n}\text{.}$ Then we show how this theory can be transported to provide combinatorial constructions of simple modules for the cyclotomic Hecke algebras $H_{r,1,n}(u_1,\dots ,u_r;q)$ and $H_{r,p,n}(x_0,\dots ,x_{d-1};q)\text{.}$ This approach shows that Jucys-Murphy type elements in the cyclotomic Hecke algebras arise naturally as images of the elements $X^{{\epsilon }_i}$ in the affine Hecke algebra. The standard Jucys-Murphy type theorems then follow almost immediately from standard affine Hecke algebra facts.

3.1 Skew shapes and standard tableaux

A partition $\lambda$ is a collection of $n$ boxes in a corner. We shall conform to the conventions in [Mac1995] and assume that gravity goes up and to the left.

$$$$

Any partition $\lambda$ can be identified with the sequence $\lambda =({\lambda }_1\ge {\lambda }_2\ge \dots )$ where ${\lambda }_i$ is the number of boxes in row $i$ of $\lambda \text{.}$ The rows and columns are numbered in the same way as for matrices. In the example above we have $\lambda =\left(553311\right)\text{.}$ If $\lambda$ and $\mu$ are partitions such that ${\mu }_i\le {\lambda }_i$ for all $i$ we write $\mu \subseteq \lambda \text{.}$ The skew shape $\lambda /\mu$ consists of all boxes of $\lambda$ which are not in $\mu \text{.}$ Any skew shape is a union of connected components. Number the boxes of each skew shape $\lambda /\mu$ along major diagonals from southwest to northeast and

$$\text{write}\hspace{0.17em}{\text{box}}_i\hspace{0.17em}\text{to indicate the box numbered}\hspace{0.17em}i\text{.}$$

Let $\lambda /\mu$ be a skew shape with $n$ boxes. A standard tableau of shape $\lambda /\mu$ is a filling of the boxes in the skew shape $\lambda /\mu$ with the numbers $1,\dots ,n$ such that the numbers increase from left to right in each row and from top to bottom down each column.

3.2 Placed skew shapes

Let $ℝ+i[0,2\pi /\text{ln}\left(q^2\right))=\{a+bi\hspace{0.17em}\mid \hspace{0.17em}a\in ℝ,0\le b\le 2\pi /\text{ln}\left(q^2\right)\}\subseteq ℂ\text{.}$ If $q$ is a positive real number then the function

$$\begin{array}{ccc}ℝ+i[0,2\pi /\text{ln}\left(q^2\right))& ⟶& {ℂ}^{*}\\ x& ⟼& q^{2x}=e^{\text{ln}\left(q^2\right)x}\end{array}$$

is a bijection. The elements of $[0,1)+i[0,2\pi /\text{ln}\left(q^2\right))$ index the $\mathbb{Z}\text{-cosets}$ in $ℝ+i[0,2\pi /\text{ln}\left(q^2\right))\text{.}$

A placed skew shape is a pair $(c,\lambda /\mu )$ consisting of a skew shape $\lambda /\mu$ and a content function

$$\begin{array}{c}\begin{array}{cc}c:\{\text{boxes of}\hspace{0.17em}\lambda /\mu \}⟶ℝ+i[0,2\pi /\text{ln}\left(q^2\right))\text{such that}& \text{(3.3)}\end{array}\\ \begin{array}{cc}c\left({\text{box}}_i\right)-c\left({\text{box}}_i\right)\ge 0& \text{if}\hspace{0.17em}i<j\hspace{0.17em}\text{and}\hspace{0.17em}c\left({\text{box}}_j\right)-c\left({\text{box}}_i\right)\in \mathbb{Z},\\ c\left({\text{box}}_j\right)=c\left({\text{box}}_i\right)+1,& \text{if}\hspace{0.17em}{\text{box}}_i\hspace{0.17em}\text{and}\hspace{0.17em}{\text{box}}_j\hspace{0.17em}\text{are on adjacent diagonals, and}\\ c\left({\text{box}}_i\right)=c\left({\text{box}}_j\right),& \text{if}\hspace{0.17em}{\text{box}}_i\hspace{0.17em}\text{and}\hspace{0.17em}{\text{box}}_j\hspace{0.17em}\text{are on the same diagonal.}\end{array}\end{array}$$

This is a generalization of the usual notion of the content of a box in a partition (see [Mac1995] I §1 Ex. 3).

Suppose that $(c,\lambda /\mu )$ is a placed skew shape such that $c$ takes values in $\mathbb{Z}\text{.}$ One can visualize $(c,\lambda /\mu )$ by placing $\lambda /\mu$ on a piece of infinite graph paper where the diagonals of the graph paper are indexed consecutively (with elements of $\mathbb{Z}\text{)}$ from southeast to northwest. The content of a box $b$ is the index $c\left(b\right)$ of the diagonal that $b$ is on. In the general case, when $c$ takes values in $ℝ+i[0,2\pi /\text{ln}\left(q^2\right)),$ one imagines a book where each page is a sheet of infinite graph paper with the diagonals indexed consecutively (with elements of $\mathbb{Z}\text{)}$ from southeast to northwest. The pages are numbered by values $\beta \in [0,1)+i[0,2\pi /\text{ln}\left(q^2\right))$ and there is a skew shape ${\lambda }^{\left(\beta \right)}/{\mu }^{\left(\beta \right)}$ placed on page $\beta \text{.}$ The skew shape $\lambda /\mu$ is a union of the disjoint skew shapes on each page,

$$\begin{array}{cc}\lambda /\mu \underset{\beta }{⨆}({\lambda }^{\left(\beta \right)}/{\mu }^{\left(\beta \right)}),\beta \in [0,1)+i[0,2\pi /\text{ln}\left(q^2\right)),& \text{(3.4)}\end{array}$$

and the content function is given by

$$\begin{array}{cc}c\left(b\right)=\text{(page number of the page containing}\hspace{0.17em}b\text{)}+\text{(index of the diagonal containing}\hspace{0.17em}b\text{)}\text{.}& \text{(3.5)}\end{array}$$

for a box $b\in \lambda /\mu \text{.}$

3.6 Example

The following diagrams illustrate standard tableaux and the numbering of boxes in a skew shape $\lambda /\mu \text{.}$

$$\begin{array}{cc} 10 12 13 14 6 8 11 5 7 9 4 2 3 1 & 3 4 9 12 1 5 10 7 13 14 2 6 8 11 \\ \lambda /\mu \hspace{0.17em}\text{with boxes numbered}& \text{A standard tableau}\hspace{0.17em}L\hspace{0.17em}\text{of shape}\hspace{0.17em}\lambda /\mu \end{array}$$

The following picture shows the contents of the boxes in the placed skew shape $(c,\lambda /\mu )$ such that the sequence

$$(c\left({\text{box}}_1\right),\dots ,c\left({\text{box}}_n\right))\hspace{0.17em}\text{is}\hspace{0.17em}(-7,-6,-5,-2,0,1,1,2,2,3,3,4,5,6)\text{.}$$ $$\begin{array}{c} 3 4 5 6 1 2 3 0 1 2 -2 -6 -5 -7 \\ \text{Contents of the boxes of}\hspace{0.17em}(c,\lambda /\mu )\end{array}$$

The following picture shows the contents of the boxes in the placed skew shape $(c,\lambda /\mu )$ such that

$$(c\left({\text{box}}_1\right),\dots ,c\left({\text{box}}_n\right))=(-7,-6,-5,-\frac{3}{2},\frac{1}{2},\frac{3}{2},\frac{3}{2},\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2},\frac{9}{2},\frac{11}{2},\frac{13}{2})\text{.}$$

This “book” has two pages, with page numbers 0 and $\frac{1}{2}\text{.}$

3.7 Calibrated $H_{\infty ,1,n}\text{-modules}$

A finite dimensional ${\stackrel{\sim }{H}}_{\infty ,1,n}\text{-modules}$ $M$ is calibrated if it has a basis $\left\{v_t\right\}$ such that for each $\lambda \in L$ and each $v_t$ in the basis

$$X^{\lambda }{v}_t=t\left(X^{\lambda }\right)v_t,\text{for some}\hspace{0.17em}t\left(X^{\lambda }\right)\in ℂ\text{.}$$

This is the class of representations of the affine Hecke algebra for which there is a good theory of Young tableaux [Ram2003].

The following theorem classifies and constructs all irreducible calibrated representations of the affine Hecke algebra $H_{\infty ,1,n}\text{.}$ The construction is a direct generalization of A. Young’s classical “seminormal construction” of the irreducible representations of the symmetric group [You1931]. Young’s construction was generalized to Iwahori-Hecke algebras of type A by Hoefsmit [Hoe1974] and Wenzl [Wen1988] independently, to Iwahori-Hecke algebras of types B and D by Hoefsmit [Hoe1974] and to cyclotomic Hecke algebras by Ariki and Koike [AKo1994]. In (3.7) and (3.11) below we show how all of these earlier generalizations of Young’s construction can be obtained from Theorem 3.8. Some parts of Theorem 3.8 are originally due to I. Cherednik, and are stated in [Che1987, §3].

Theorem 3.8. ([Ram1997, Theorem 4.1]) Let $(c,\lambda /\mu )$ be a placed skew shape with n boxes. Define an action of $H_{\infty ,1,n}$ on the vector space

$${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}=ℂ\text{-span}\{v_L\hspace{0.17em}\mid \hspace{0.17em}L\hspace{0.17em}\text{is a standard tableau of shape}\hspace{0.17em}\lambda /\mu \}$$

by the formulas

$$\begin{array}{ccc}{X}^{{\epsilon }_i}{v}_L& =& q^{2c\left(L\left(i\right)\right)}{v}_L,\\ T_i{v}_L& =& {\left(T_i\right)}_{LL}{v}_L+(q^{-1}+{\left(T_i\right)}_{LL})v_{s_{i}L},\end{array}$$

where $s_{i}L$ is the same as $L$ except that the entries $i-1$ and $i$ are interchanged,

$$\begin{array}{ccc}{\left(T_i\right)}_{LL}& =& \frac{q-q^{-1}}{1-{q^2}^{(c\left(L(i-1)\right)-c\left(L\left(i\right)\right))}},\\ v_{s_{i}L}& =& 0,\text{if}\hspace{0.17em}{s}_{i}L\hspace{0.17em}\text{is not a standard tableau,}\end{array}$$

and $L\left(i\right)$ denotes the box of $L$ containing the entry $i\text{.}$

1. ${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}$ is a calibrated irreducible $H_{\infty ,1,n}\text{-module.}$
2. The modules ${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}$ are non-isomorphic.
3. Every irreducible calibrated $H_{\infty ,1,n}\text{-module}$ is isomorphic to ${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}$ for some placed skew shape $(c,\lambda /\mu )\text{.}$

Remark 3.9. All of the irreducible modules for the affine Hecke algebra have been classified and constructed by Kazhdan and Lusztig [KLu0862716]. The construction in [KLu0862716] is geometric and noncombinatorial. It is nontrivial (but not very difficult) to relate the construction of Theorem 3.8 and the classification in [KLu0862716].

3.10 Calibrated $H_{\infty ,p,n}\text{-modules}$

A finite dimensional $H_{\infty ,p,n}$ module $M$ is calibrated if it has a basis $\left\{v_t\right\}$ such that for each $\lambda \in L_p$ and each $v_t$ in the basis

$$X^{\lambda }{v}_t=t\left(X^{\lambda }\right)v_t,\text{for some}\hspace{0.17em}t\left(X^{\lambda }\right)\in ℂ\text{.}$$

Let us show how Theorem 3.8, Theorem 2.2 and Theorem A.13 provide explicit constructions of simple calibrated $H_{\infty ,p,n}\text{-modules.}$ The resulting construction is a generalization of the construction of $H_{r,p,n}(x_0,\dots ,x_{d-1};q)\text{-modules}$ given by Ariki [Ari1995] (as amplified and applied in [HRa1998]). Comparing the following machinations with those in [HRa1998, §3] (where more pictures are given) will be helpful.

The $(\mathbb{Z}/p\mathbb{Z})\text{-action}$ on $H_{\infty ,1,n}$ induces an action of $\mathbb{Z}/p\mathbb{Z}$ on the simple $H_{\infty ,1,n}\text{-modules,}$ as in (A.1) of the Appendix, and this action takes calibrated modules to calibrated modules since the $(\mathbb{Z}/p\mathbb{Z})\text{-action}$ on $H_{\infty ,1,n}$ preserves the subalgebra $ℂ\left[X\right]\text{.}$ The $\mathbb{Z}/p\mathbb{Z}$ action on simple calibrated modules can be described combinatorially as follows.

If $(c,\lambda /\mu )$ is a placed skew shape with $n$ boxes and $g\in \mathbb{Z}/p\mathbb{Z}$ define

$$\begin{array}{cc}g(c,\lambda /\mu )=(c-i\alpha /p,\lambda /\mu ),\text{where}\hspace{0.17em}\alpha =2\pi /\text{ln}\left(q^2\right),& \text{(3.11)}\end{array}$$

and $c-i\alpha /p$ denotes the content function defined by $(c-i\alpha /p)\left(b\right)=c\left(b\right)-i\alpha /p,$ for all boxes $b\in \lambda /\mu \text{.}$ To make this definition we are identifying the set $[0,\alpha )$ with $ℝ/\alpha \mathbb{Z}\text{.}$ One can imagine the placed skew shape as a book with pages numbered by values $\beta \in [0,1)+i(ℝ/\alpha \mathbb{Z})$ and a skew shape ${\lambda }^{\left(\beta \right)}/{\mu }^{\left(\beta \right)}$ on each page. The action of

$$g\hspace{0.17em}\text{cyclically permutes the pages numbered}\hspace{0.17em}\beta +i(k/p)\alpha ,0\le k<p\text{.}$$

If $L$ is a standard tableau of shape $(c,\lambda /\mu )$ let $gL$ denote the same filling of $\lambda /\mu$ as $L$ but viewed as a standard tableaux of shape $g(c,\lambda /\mu )\text{.}$

Let ${}^g{\stackrel{\sim }{H}}^{(c,\lambda /\mu )}$ be the $H_{\infty ,1,n}\text{-module}$ ${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}$ except twisted by the automorphism $g,$ see (A.1) in the Appendix. It follows from the formulas in Theorem 3.8 that the map

$$\begin{array}{cc}\begin{array}{cccc}\varphi :& {}^g{\stackrel{\sim }{H}}^{(c,\lambda /\mu )}& ⟶& {\stackrel{\sim }{H}}^{g(c,\lambda /\mu )}\\ & v_L& ⟼& v_{gL}\end{array}& \text{(3.12)}\end{array}$$

is an $H_{\infty ,1,n}\text{-module}$ isomorphism. Indeed, since $g^{-1}\left(T_j\right)=T_j$ and ${\left(T_j\right)}_{gL,gL}={\left(T_j\right)}_{LL},$

$$\begin{array}{ccc}\varphi (T_j\circ v_L)& =& T_j\varphi \left(v_L\right),\text{and}\\ \phi (X^{{\epsilon }_j}\circ v_L)& =& \varphi \left(g^{-1}\left(X^{{\epsilon }_j}\right)v_L\right)\\ & =& \varphi \left({$ \xi $}^{-1}{X}^{{\epsilon }_j}{v}_L\right)\\ & =& {q^2}^{(c\left(L\left(j\right)\right)-i\alpha /p)}{v}_{gL}\\ & =& X^{{\epsilon }_j}{v}_{gL}\\ & =& X^{{\epsilon }_j}\varphi \left(v_L\right),\end{array}$$

where $\circ$ denotes the $H_{\infty ,1,n}\text{-action}$ on ${}^g{\stackrel{\sim }{H}}^{(c,\lambda /\mu )}$ as in (A.1). Identify ${}^g{\stackrel{\sim }{H}}^{(c,\lambda /\mu )}$ with ${\stackrel{\sim }{H}}^{g(c,\lambda /\mu )}$ via the isomorphism in (3.12).

Let $(c,\lambda /\mu )$ be a placed skew shape with $n$ boxes and let $K_{(c,\lambda /\mu )}$ be the stabilizer of $(c,\lambda /\mu )$ under the action of $\mathbb{Z}/p\mathbb{Z}\text{.}$ The cyclic group $K_{(c,\lambda /\mu )}$ is a realization of the inertia group of ${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}$ and

$$K_{(c,\lambda /\mu )}=\{{\left(g^{\kappa }\right)}^{\ell }:{\stackrel{\sim }{H}}^{(c,\lambda /\mu )}\to {\stackrel{\sim }{H}}^{(c,\lambda /\mu )}\hspace{0.17em}\mid \hspace{0.17em}0\le \ell \le \mid K_{(c,\lambda /\mu )}\mid -1\},$$

where $\kappa$ is the smallest integer between 1 and $p$ such that $g^{\kappa }(c,\lambda /\mu )=(c,\lambda /\mu )$ and $\mid K_{(c,\lambda /\mu )}\mid$ is the order of $K_{(c,\lambda /\mu )}\text{.}$ The elements of $K_{(c,\lambda /\mu )}$ are all $H_{\infty ,p,n}\text{-module}$ isomorphisms. Since $K_{(c,\lambda /\mu )}$ is a cyclic group the irreducible $K_{(c,\lambda /\mu )}\text{-modules}$ are all one-dimensional and the characters of these modules are given explicitly by

$$\begin{array}{cccc}{\eta }_j:& K_{(c,\lambda /\mu )}& ⟶& ℂ\\ & g^{\kappa }& ⟼& {\xi }^{j\kappa },\end{array}0\le j\le \mid K_{(c,\lambda /\mu )}\mid -1,$$

since ${\xi }^{\kappa }$ is a primitive $\mid K_{(c,\lambda /\mu )}\mid \text{-th}$ root of unity. The element

$$\begin{array}{cc}{p}_j=\sum _{\ell =0}^{\mid K_{(c,\lambda /\mu )}\mid -1}{\xi }^{-j\ell \kappa }{g}^{\ell \kappa }\text{.}& \text{(3.13)}\end{array}$$

is the minimal idempotent of the group algebra $ℂK_{(c,\lambda /\mu )}$ corresponding to the irreducible character ${\eta }_j\text{.}$ It follows (from a standard double centralizer result, [Bou1958]) that, as an $(H_{\infty ,p,n},K_{(c,\lambda /\mu )})\text{-bimodule,}$

$$\begin{array}{cc}{\stackrel{\sim }{H}}^{(c,\lambda /\mu )}\cong \underset{j=0}{\overset{\mid K_{(c,\lambda /\mu )}\mid -1}{⨁}}{\stackrel{\sim }{H}}^{(c,\lambda /\mu ,j)}\otimes K^{\left(j\right)},\text{where}{\stackrel{\sim }{H}}^{(c,\lambda /\mu ,j)}=p_j{\stackrel{\sim }{H}}^{(c,\lambda /\mu )},& \text{(3.14)}\end{array}$$

and $K^{\left(j\right)}$ is the irreducible $K_{(c,\lambda /\mu )}\text{-module}$ with character ${\eta }_j\text{.}$ The following theorem now follows from Theorem A.13 of the Appendix.

Theorem 3.15 Let $(c,\lambda /\mu )$ be a placed skew shape with $n$ boxes and let ${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}$ be the simple calibrated $H_{\infty ,1,n}\text{-module}$ constructed in Theorem 3.8. Let $K_{(c,\lambda /\mu )}$ be the inertia group of ${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}$ corresponding to the action of $\mathbb{Z}/p\mathbb{Z}$ on $H_{\infty ,1,n}$ defined by (A.1). If $p_j$ is the minimal idempotent of $K_{(c,\lambda /\mu )}$ given by (3.13) then

$${\stackrel{\sim }{H}}^{(c,\lambda /\mu ,j)}=p_j{\stackrel{\sim }{H}}^{(c,\lambda /\mu )}$$

is a simple calibrated $H_{\infty ,p,n}\text{-module.}$

Theorem 3.15 provides a generalization of the construction of the $H_{r,p,n}\text{-modules}$ which was given by Ariki [Ari1995] and extended and applied in [HRa1998].

3.16 Simple $H_{r,1,n}(u_1,\dots ,u_r;q)\text{-modules}$

Many (usually all) of the simple $H_{r,1,n}(u_1,\dots ,u_r;q)\text{-modules}$ can be constructed with Theorems 2.8 and 3.8.

If $\lambda /\mu$ is a skew shape define

$$NW(\lambda /\mu )=\{\text{northwest corner boxes of}\hspace{0.17em}\lambda /\mu \},$$

so that $NW(\lambda /\mu )$ is the set of boxes $b\in \lambda /\mu$ such that there is no box of $\lambda /\mu$ immediately above or immediately to the left of $b\text{.}$

Theorem 3.17. Fix $u_1,\dots ,u_r\in {ℂ}^{*}$ and let $(c,\lambda /\mu )$ be a placed skew shape with $n$ boxes. If

$$\{q^{2c\left(b\right)}\hspace{0.17em}\mid \hspace{0.17em}b\in NW(\lambda /\mu )\}\subseteq \{u_1,\dots ,u_r\}$$

then the $H_{\infty ,1,n}\text{-module}$ ${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}$ is a simple $H_{r,1,n}(u_1,\dots ,u_r;q)\text{-module}$ (via Theorem 2.8).

		Proof.
	Let $(c,\lambda /\mu )$ be a placed skew shape with $n$ boxes and let ${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}$ be the simple $H_{\infty ,1,n}\text{-module}$ of Theorem 3.8. Let ${\rho }^{(c,\lambda /\mu )}:H_{\infty ,1,n}\to \text{End}\left({\stackrel{\sim }{H}}^{(c,\lambda /\mu )}\right)$ be the representation corresponding to ${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}\text{.}$ By the formulas in Theorem 3.8 the matrix $\rho \left(X^{{\epsilon }_1}\right)$ is diagonal with eigenvalues $q^{2c\left(L\left(1\right)\right)},$ for $L\in {ℱ}^{(c,\lambda /\mu )},$ where ${ℱ}^{(c,\lambda /\mu )}$ is the set of standard tableaux of shape $\lambda /\mu \text{.}$ The boxes $L\left(1\right),$ $L\in {ℱ}^{(c,\lambda /\mu )}$ are exactly the northwest corner boxes of $\lambda /\mu$ and so the minimal polynomial of ${\rho }^{\left(X^{{\epsilon }_1}\right)}$ is $$p\left(t\right)=\prod _{b\in NW(\lambda /\mu )}(t-q^{2c\left(b\right)})\text{.}$$ Thus the condition $\{q^{2c\left(b\right)}\hspace{0.17em}\mid \hspace{0.17em}b\in NW(\lambda /\mu )\}\subseteq \{u_1,\dots ,u_r\}$ is exactly what is needed for ${\stackrel{\sim }{H}}^{(c,\lambda /\mu )}$ to be an $H_{r,1,n}(u_1,\dots ,u_r;q)\text{-module}$ via Theorem 2.8. $\square$

Theorem 3.18. If the cyclotomic Hecke algebra $H_{r,1,n}(u_1,u_2,\dots ,u_r;q)$ is semisimple, then its simple modules are the modules ${\stackrel{\sim }{H}}^{(c,\lambda )}$ constructed in Theorem 3.17, where $\lambda =({\lambda }^{\left(1\right)},\dots ,{\lambda }^{\left(r\right)})$ is an $r\text{-tuple}$ of partitions with a total of $n$ boxes and $c$ is the content function determined by

$$q^{2c\left(b\right)}=u_i,\text{if}\hspace{0.17em}b\hspace{0.17em}\text{is the northwest corner box of}\hspace{0.17em}{\lambda }^{\left(i\right)}\text{.}$$

		Proof.
	The equations $q^{2c\left(b\right)}=u_i,$ for $b\in NW\left({\lambda }^{\left(i\right)}\right),$ determine the values $c\left(b\right)$ for all boxes $b\in \lambda$ and thus the pair $(c,\lambda )$ defines a placed skew shape. By a Theorem of Ariki [Ari1994], $H_{r,1,n}(u_1,\dots ,u_r;q)$ is semisimple if and only if $${\left[n\right]}_q!\ne 0\text{and}{u}_i{u}_j^{-1}\notin \{1,q^2,\dots ,q^{2n}\},$$ where ${\left[n\right]}_q!={\left[n\right]}_q{[n-1]}_q\dots {\left[2\right]}_q{\left[1\right]}_q$ and ${\left[k\right]}_q=(q^k-q^{-k})/(q-q^{-1})\text{.}$ These conditions guarantee that $(c,\lambda )$ is a placed skew shape and that the ${\stackrel{\sim }{H}}_{\infty ,1,n}\text{-module}$ ${\stackrel{\sim }{H}}^{(c,\lambda )}$ defined in Theorem 3.8 is well defined and irreducible. The reduction in Theorem 2.8 makes ${\stackrel{\sim }{H}}^{(c,\lambda )}$ into a simple $H_{r,1,n}(u_1,\dots ,u_r;q)\text{-module}$ and a count of standard tableaux (using binomial coefficients and the classical identity for the symmetric group case) shows that $$\sum _{\lambda =({\lambda }^{\left(1\right)},\dots ,{\lambda }^{\left(r\right)})}\text{dim}{\left({\stackrel{\sim }{H}}^{(c,\lambda )}\right)}^2=r^{n}n!=\text{dim}\left(H_{r,1,n}\right),$$ where the sum is over all $r\text{-tuples}$ $\lambda =({\lambda }^{\left(1\right)},\dots ,{\lambda }^{\left(r\right)})$ with a total of $n$ boxes. Thus the ${\stackrel{\sim }{H}}^{(c,\lambda )}$ are a complete set of simple $H_{r,1,n}(u_1,\dots ,u_r;q)\text{-modules.}$ $\square$

Theorem 3.18 demonstrates that the construction of simple modules for $H_{r,1,n}$ by Ariki and Koike [AKo1994, Theorem 3.7], for $H_{2,1,n}(p,-p^{-1};q)$ by Hoefsmit [Hoe1974], for $H_{1,1,n}(1;q)$ by Hoefsmit [Hoe1974] and Wenzl [Wen1988] (independently), and for $ℂS_n=H_{1,1,n}(1;1)$ and $ℂW{B}_n=H_{2,1,n}(1,1;1)$ by Young [You1929,You1931], are all special cases of Theorem 3.8.

3.19 Jucys-Murphy elements in cyclotomic Hecke algebras

The following result is well known, but we give a new proof which shows that the cyclotomic Hecke algebra analogues of the Jucys-Murphy elements which have appeared in the literature (see [BMM1993], [Ram1997], [DJM1995] and the references there) come naturally from the affine Hecke algebra $H_{\infty ,1,n}\text{.}$

Corollary 3.20. Let $H_{r,1,n}(u_1,\dots ,u_r;q)$ and $H_{r,p,n}(x_0,\dots ,x_{d-1};q)$ be the cyclotomic Hecke algebras defined in (1.1) and (1.2).

1. The elements $$M_i=T_i\dots T_2{T}_1{T}_2\dots T_i,1\le i\le n,$$ generate a commutative subalgebra of $H_{r,1,n}(u_1,\dots ,u_r;q)\text{.}$
2. If $H_{r,1,n}(u_1,\dots ,u_r;q)$ is semisimple then every simple $H_{r,1,n}(u_1,\dots ,u_r;q)\text{-module}$ has a basis of simultaneous eigenvectors of the elements $M_i\text{.}$
3. The elements $$M_1^p=a_0,\text{and}{M}_i{M}_1^{-1}=a_i\dots a_3{a}_2{a}_1{a}_3\dots a_i,2\le i\le n,$$ generate a commutative subalgebra of $H_{r,p,n}(x_0,\dots ,x_{d-1};q)\text{.}$
4. If $H_{r,p,n}(x_0,\dots ,x_{d-1};q)$ is semisimple then every simple $H_{r,p,n}(x_0,\dots ,x_{d-1};q)\text{-module}$ has a basis of simultaneous eigenvectors of the elements $M_1^p$ and $M_i{M}_1^{-1},$ $2\le i\le n\text{.}$

		Proof.
	(a) The elements $X^{{\epsilon }_i},$ $1\le i\le n,$ generate the subalgebra $ℂ\left[X\right]\subseteq H_{\infty ,1,n}\text{.}$ Inductive use of the relation (1.4) shows that $M_i=\Phi \left(X^{{\epsilon }_i}\right),$ where $\Phi :H_{\infty ,1,n}\to H_{r,1,n}(u_1,\dots ,u_r;q)$ is the homomorphism in Proposition 2.5. Thus the subalgebra of $H_{r,1,n}(u_1,\dots ,u_r;q)$ generated by the element $M_i$ is the image of $ℂ\left[X\right]$ under the homomorphism $\Phi \text{.}$ (b) is an immediate consequence of Theorem 3.18, the construction described in Theorem 3.8, and the fact that $M_i=\Phi \left(X^{{\epsilon }_i}\right)\text{.}$ (c)The elements $X^{p{\epsilon }_1}$ and $X^{{\epsilon }_i-{\epsilon }_1},$ $2\le i\le n,$ generate the subalgebra $ℂ\left[X^{L_p}\right]\subseteq H_{\infty ,p,n}$ and the images of these elements under the homomorphism ${\Phi }_p:H_{\infty ,p,n}\to H_{r,p,n}(x_0,\dots ,x_{d-1};q)$ are the elements $M_1^p$ and $M_i{M}_1^{-1}\text{.}$ The proof of part (d) uses the construction described in Theorem 3.15 and is analogous to the proof of part (b). $\square$

(3.21) The center of $H_{r,1,n}(u_1,\dots ,u_r;q)$

It is an immediate consequence of (1.12) and the proof of Corollary 3.20 that

$$ℂ{[M_1,\dots ,M_n]}^{S_n}\subseteq Z\left(H_{r,1,n}(u_1,\dots ,u_r;q)\right),$$

where $M_i=T_i{T}_{i-1}\dots T_2{T}_1{T}_2\dots T_{i-1}{T}_i\text{.}$ The following proposition shows that this inclusion is an equality.

Proposition 3.22. If the cyclotomic Hecke algebra $H_{r,1,n}(u_1,\dots ,u_r;q)$ is semisimple then its center

$$Z\left(H_{r,1,n}(u_1,\dots ,u_r;q)\right)=ℂ{[M_1,\dots ,M_n]}^{S_n},$$

where $M_i=T_i{T}_{i-1}\dots T_2{T}_1{T}_2\dots T_{i-1}{T}_i$ and $ℂ{[M_1,\dots ,M_n]}^{S_n}$ is the ring of symmetric polynomials in $M_1,\dots ,M_n\text{.}$

		Proof.
	By (1.12) $Z\left(H_{\infty ,1,n}\right)=ℂ{\left[X\right]}^{S_n}\text{.}$ Thus, since $M_i=\Phi \left(X^{{\epsilon }_i}\right),$ where $\Phi :H_{\infty ,1,n}\to H_{r,1,n}(u_1,\dots ,u_r;q)$ is the surjective homomorphism of Proposition 2.5, it follows that $$ℂ{[M_1,\dots ,M_n]}^{S_n}=\Phi \left(ℂ{\left[X\right]}^{S_n}\right)=\Phi \left(Z\left(H_{\infty ,1,n}\right)\right)\subseteq Z\left(H_{r,1,n}(u_1,\dots ,u_r;q)\right)\text{.}$$ For the reverse inclusion we need to show that the action of the elements $ℂ{[M_1,\dots ,M_n]}^{S_n}$ distinguishes the simple $H_{r,1,n}(u_1,\dots ,u_r;q)\text{-modules.}$ Let $\lambda =({\lambda }^{\left(1\right)},\dots ,{\lambda }^{\left(r\right)})$ be an $r\text{-tuple}$ of partitions with a total of $n$ boxes and let ${\stackrel{\sim }{H}}^{(c,\lambda )}$ be the corresponding simple $H_{r,1,n}(u_1,\dots ,u_r;q)\text{-module}$ as constructed by Theorem 3.18 (and Theorem 3.8). If $L$ is a standard tableau of shape $\lambda$ then $M_1,\dots ,M_n$ act on $v_L$ by the multiset of values $(q^{2c\left(L\left(1\right)\right)},\dots ,q^{2c\left(L\left(n\right)\right)})\text{.}$ The elementary symmetric functions $e_i(M_1,\dots ,M_n)$ act on ${\stackrel{\sim }{H}}^{(c,\lambda )}$ by the values $a_i=e_i(q^{2c\left(L\left(1\right)\right)},\dots ,q^{2c\left(L\left(n\right)\right)})\text{.}$ Note that $a_i$ does not depend on the choice of the standard tableau $L$ (since $e_i(M_1,\dots ,M_n)\in Z\left(H_{r,1,n}\right)\text{).}$ We show that the simple module ${\stackrel{\sim }{H}}^{(c,\lambda )}$ is determined by the values $a_1,\dots ,a_n\text{.}$ This shows that the simple modules are distinguished by the elements of $ℂ{[M_1,\dots ,M_n]}^{S_n}\text{.}$ Let us explain how the values $a_1,\dots ,a_n$ determine the placed skew shape $(c,\lambda )\text{.}$ There is a unique (unordered) multiset of values $b_1,\dots ,b_n$ such that $a_i=e_i(b_1,\dots ,b_n)$ for all $1\le i\le n\text{.}$ The $b_i$ are determined by the equation $$(t-b_1)\dots (t-b_n)=t^n-a_1{t}^{n-1}+a_2{t}^{n-2}-a_3{t}^{n-3}+\dots \pm a_n\text{.}$$ From the previous paragraph, the multiset $\{b_1,\dots ,b_n\}$ must be the same as the multiset $\{q^{2c\left(L\left(1\right)\right)},\dots ,q^{2c\left(L\left(n\right)\right)}\}\text{.}$ In this way the values $a_1,\dots ,a_n$ determine the multiset $S=\{c\left(L\left(1\right)\right),\dots ,c\left(L\left(n\right)\right)\}\text{.}$ The multiset $S$ is a disjoint union $S=S_1\bigsqcup \dots \bigsqcup S_r$ of multisets such that each $S_i$ is in a single $\mathbb{Z}\text{-coset}$ of $ℝ+i[0,2\pi /\text{ln}\left(q^2\right))i$ (see (3.4)). These $\mathbb{Z}$ cosets are determined by the values $u_1,\dots ,u_r$ and thus there is a one-to-one correspondence between the $u_i$ and the multisets $S_i\text{.}$ Then $\mid {\lambda }^{\left(i\right)}\mid =\text{Card}\left(S_i\right),$ the nonempty diagonals of ${\lambda }^{\left(i\right)}$ are determined by the values in $S_i,$ and the lengths of the diagonals of ${\lambda }^{\left(i\right)}$ are the multiplicities of the values of the elements of $S_i\text{.}$ This information completely determines ${\lambda }^{\left(i\right)}$ for each $i\text{.}$ Thus $(c,\lambda )$ is determined by the values $a_1,\dots ,a_n\text{.}$ $\square$

Remark 3.23. In the language of affine Hecke algebra representations (see [Ram2003]) the proof of Proposition 3.22 shows that (when $H_{r,1,n}(u_1,\dots ,u_r;q)$ is semisimple) the simple $H_{r,1,n}(u_1,\dots ,u_r;q)\text{-modules}$ ${\stackrel{\sim }{H}}^{(c,\lambda )}$ all have different central characters (as $H_{\infty ,1,n}$ modules).

Remark 3.24. The elements $M_1^p$ and $M_i{M}_1^{-1}$ from Corollary 3.20 cannot be used to obtain a direct analogue of Proposition 3.22 for $H_{r,p,n}(x_0,\dots ,x_{d-1};q)\text{.}$ This is because all of the $H_{\infty ,p,n}\text{-modules}$ $V_j$ appearing in the decomposition (3.14) will have the same central character. However, when $n$ is odd, the natural analogue of Proposition 3.22 does hold for Iwahori-Hecke algebras of type $D_n,$ $H{D}_n\left(q\right)=H_{2,2,n}(1;q)\text{.}$ In that case, every simple $H_{2,2,n}(1;q)\text{.}$ has trivial inertia group and the decomposition in (3.14) has only one summand.

Notes and References

This is an excerpt of the paper entitled Affine Hecke Algebras, Cyclotomic Hecke algebras and Clifford Theory authored by Arun Ram and Jacqui Ramagge. It was dedicated to Professor C.S. Seshadri on the occasion of his 70th birthday.

Research partially supported by the Natioanl Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).

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