Representations of Hecke Algebras of Finite Groups with BN-Pairs of Classical Type

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

Last update: 8 June 2014

Notes and References

This is a typed copy of Peter Norbert Hoefsmit's thesis Representations of Hecke Algebras of Finite Groups with BN-Pairs of Classical Type published in August, 1974.

Chapter 1.$$Preliminaries

Partitions and Tableaux

A partition of $n$ is an ordered set of positive numbers $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_k)$ such that $$n={\alpha }_1+\dots +{\alpha }_k,{\alpha }_1\ge {\alpha }_2\ge \dots \ge {\alpha }_k\text{,}$$ $k$ arbitrary. Such a partition $\left(\alpha \right)$ is said to have $k$ parts and its length, $\left|\alpha \right|,$ is $n\text{.}$ Thus $(4,3,3,1)$ is a partition of $11\text{.}$ It may also be written $(4,3^2,1)$ and a similar notation will be used elsewhere.

We represent $\left(\alpha \right)$ by a Young diagram, $D\left(\alpha \right)\text{,}$ having ${\alpha }_1$ squares in the first row, ${\alpha }_2$ squares in the second row and so on, the $j\text{-th}$ squares of the rows making a column. $D\left(\alpha \right)$ is said to have shape $\left(\alpha \right)\text{.}$ Thus $$\begin{array}{c} \end{array}$$ is the Young diagram of shape $(4,3^2,1)\text{.}$ The square in the $i\text{-th}$ row and $j\text{-th}$ column of $D\left(\alpha \right)$ is said to have coordinates $(i,j)$ and is called the $(i,j)\text{-square}\text{.}$

Let ${\alpha }_i^{\prime }$ denote the number of squares appearing in the $i\text{-th}$ column of $D\left(\alpha \right)\text{.}$ The Young diagram $D\left(\alpha \prime \right),$ obtained by interchanging the rows and columns of $D\left(\alpha \right)\text{,}$ is called the conjugate of $D\left(\alpha \right)$ and the partition $\left(\alpha \prime \right)=({\alpha }_1^{\prime },\dots ,{\alpha }_s^{\prime })$ is called the conjugate of $\left(\alpha \right)\text{.}$

The $n$ letters $1,\dots ,n$ may be arranged in the squares of $D\left(\alpha \right)$ in $n!$ ways. Each such arrangement is called a tableau of shape $\left(\alpha \right)\text{.}$ A tableau is called a standard tableau if the letters in every row increase from left to right and in every column from top to bottom. Thus the tableaux (ii) and (iii) are standard while (i) is not. $$\begin{array}{ccc}\begin{array}{c} 1 4 5 2 3 \end{array}& \begin{array}{c} 1 3 5 2 4 \end{array}& \begin{array}{c} 1 2 3 4 5 \end{array}\\ \text{(i)}& \text{(ii)}& \text{(iii)}\\ \multicolumn{3}{c}{\text{Figure 1}}\end{array}$$ The tableau of shape $\left(\alpha \right)$ with the $n$ letters arranged in consecutive order in the rows, starting with the first square in the first row is called the canonical tableau of shape $\left(\alpha \right)\text{.}$ The tableau (iii) above is the canonical tableau of shape $(3,2)\text{.}$

The number, $f^{\alpha },$ of standard tableaux of shape $\left(\alpha \right)$ is determined as follows (see [Rob1961], p. 44). The $(i,j)\text{-square}$ of $D\left(\alpha \right)$ determines the $(i,j)\text{-hook}$ consisting of the $(i,j)\text{-square}$ along with the ${\alpha }_i-i$ squares to the right in the $i\text{-th}$ row and the ${\alpha }_j^{\prime }-j$ squares below in the $j\text{-th}$ column. Thus the length of the $(i,j)\text{-hook}$ is $$\begin{array}{cc}\text{(1.1.1)}& h_{ij}=({\alpha }_i-i)+({\alpha }_j^{\prime }-j)+1\text{.}\end{array}$$ Then $$\begin{array}{cc}\text{(1.1.2)}& {\displaystyle f^{\alpha }=\frac{n!}{\prod _{i,j}{h}_{ij}}\text{.}}\end{array}$$

A double partition of $n$ is an ordered pair of partitions $\left(\mu \right)=(\alpha ,\beta )$ with $\left|\alpha \right|+\left|\beta \right|=n\text{.}$ If $\left(\alpha \right)=({\alpha }_1,\dots ,{\alpha }_s)$ and $\left(\beta \right)=({\beta }_1,\dots ,{\beta }_t),$ we write the double partition $\left(\mu \right)$ as $\left(\mu \right)=({\mu }_1,\dots ,{\mu }_{s+t})$ where ${\mu }_i={\alpha }_i,$ $1\le i\le s$ and ${\mu }_j={\beta }_i$ where $j>s,$ $j=s+i\text{.}$ We allow either $\left(\alpha \right)$ or $\left(\beta \right)$ to be a partition of $n$ in the above, i.e. let $\left(0\right)$ denote the empty partition. For $\left(\alpha \right)$ a partition of $n\text{,}$ $(\left(\alpha \right),\left(0\right))$ and $(\left(0\right),\left(\alpha \right))$ are distinct double partitions of $n\text{.}$ We represent $\left(\mu \right)$ by an ordered pair of Young diagrams $D\left(\mu \right)=(D\left(\alpha \right),D\left(\beta \right)),$ called the Young diagram of shape $\left(\mu \right)\text{.}$ $D\left(\mu \right)$ is considered to have $s+t$ rows, where the $i\text{-th}$ row of $D\left(\alpha \right)$ is the $i\text{-th}$ row of $D\left(\mu \right)$ and the $j\text{-th}$ row of $D\left(\beta \right)$ is the $s+j\text{-th}$ row of $D\left(\mu \right)\text{.}$ The Young diagrams of shape $(\left(\alpha \right),\left(0\right))$ and $(\left(0\right),\left(\beta \right))$ are taken to be $(-,D\left(\alpha \right))$ and $(D\left(\alpha \right),-)\text{.}$ The squares of $D\left(\mu \right)$ are identified by their coordinates in the diagrams $D\left(\alpha \right)$ and $D\left(\beta \right)\text{.}$ Thus the square of $D\left(\mu \right)$ which is in the $i\text{-th}$ row and in the $j\text{-th}$ column of $D\left(\alpha \right)$ (resp. $D\left(\beta \right)\text{)}$ is called the $(i,j)\text{-square}$ of $D\left(\alpha \right)$ (resp. $D\left(\beta \right)\text{)}$ and has coordinates $(i,j)\text{.}$ Hence distinct squares of $D\left(\mu \right)$ can have the same coordinates (for instance, the first square in the first row of $D\left(\mu \right)$ and the first square in the $s+1\text{-st}$ row of $D\left(\mu \right)$ both have coordinates $(1,1),$ where $s$ is as above).

A tableau of shape $\left(\mu \right)=(\alpha ,\beta )$ is any arrangement of the letters $1,\dots ,n$ in $D\left(\mu \right)\text{.}$ Thus a tableau $T^{\mu }$ is an ordered pair $(T^{\alpha },T^{\beta })$ where, for complementary subsets $K$ and $L$ of $\{1,\dots ,n\}$ with $\left|K\right|=\left|\alpha \right|,$ $\left|L\right|=\left|\beta \right|,$ $T^{\alpha }$ denotes any arrangement of the letters of $K$ in $D\left(\alpha \right)$ and $T^{\left(\beta \right)}$ denotes any arrangement of the letters of $L$ in $D\left(\beta \right)\text{.}$ The tableau $T^{\mu }$ is a standard tableau if the arrangement of the letters is in increasing order in the rows and columns of both $T^{\alpha }$ and $T^{\beta }\text{.}$ Thus, Figure 2 $$\begin{array}{c}\begin{array}{c} 3 5 13 4 12 7 1 6 10 11 2 8 9 \end{array}\\ \text{Figure 2}\end{array}$$ is a standard tableau of shape $((3,2,1),(4,2,1))\text{.}$ The canonical tableau of shape $\left(\mu \right)$ is the tableau where the letters are arranged consecutively in the rows of $D\left(\mu \right)\text{,}$ starting with the first square in the first row of $D\left(\mu \right)\text{.}$ The number, $f^{\mu },$ of standard tableaux of shape $\left(\mu \right)=(\alpha ,\beta )$ is (1.1.3) fμ=(nk) fαfβ, k=|α|, where $f^{\left(0\right)}=1\text{.}$

We order the standard tableaux of a given shape as follows:

Definition (1.1.4) Let $T_1^{\mu },$ $T_2^{\mu }$ denote standard tableaux of shape $\left(\mu \right)\text{.}$ We say $T_1^{\mu }$ precedes $T_2^{\mu }$ if the letters $n,n-1,\dots ,n-r+1$ appear in the same row in both tableaux but the letter $n-r$ appears in a lower row in $T_1^{\mu }$ than in $T_2^{\mu }\text{.}$ The enumeration of the standard tableaux according to their ordering is called the last letter sequence.

Thus in the last letter sequence all tableaux which have the letter $n$ in the last row precede those which have $n$ in the second to the last row. These latter tableaux precede those which have $n$ in the third to the last row and so on. Those tableaux which have $n$ in the same row are arranged by the same scheme according to the position of the letter $n-1$ and so on. It is evident that the canonical tableau is the first tableau in this ordering.

We give an example of this ordering. For the double partition $((2,1),\left(2\right)),$ there are 20 standard tableaux. Arranged according to the last letter sequence they are, $$\begin{array}{cccccccccc}\begin{array}{c} 1 2 3 4 5 \end{array}& \begin{array}{c} 1 3 2 4 5 \end{array}& \begin{array}{c} 1 2 4 3 5 \end{array}& \begin{array}{c} 1 3 4 2 5 \end{array}& \begin{array}{c} 2 3 4 1 5 \end{array}& \begin{array}{c} 1 4 2 3 5 \end{array}& \begin{array}{c} 1 4 3 2 5 \end{array}& \begin{array}{c} 2 4 3 1 5 \end{array}& \begin{array}{c} 1 2 5 3 4 \end{array}& \begin{array}{c} 1 3 5 2 4 \end{array}\\ \begin{array}{c} 2 3 5 1 4 \end{array}& \begin{array}{c} 1 4 5 2 3 \end{array}& \begin{array}{c} 2 4 5 1 3 \end{array}& \begin{array}{c} 3 4 5 1 2 \end{array}& \begin{array}{c} 1 5 2 3 4 \end{array}& \begin{array}{c} 1 5 3 2 4 \end{array}& \begin{array}{c} 2 5 3 1 4 \end{array}& \begin{array}{c} 1 5 4 2 3 \end{array}& \begin{array}{c} 2 5 4 1 3 \end{array}& \begin{array}{c} 3 5 4 1 2 \end{array}\\ \multicolumn{10}{c}{\text{Figure 3}}\end{array}$$ Finally we define the notion of axial distance.

Definition (1.1.5) For squares $A$ and $B$ in a Young diagram $D\left(\mu \right)$ with coordinates $(i,j)$ and $(s,t)$ respectively, define the axial distance, $\rho ,$ from $A$ to $B$ to be $$\rho =(t-s)-(j-i)$$

Axial distance has a simple graphical interpretation. Suppose the squares $A$ and $B$ are in the same diagram of $D\left(\mu \right)=(D\left(\alpha \right),D\left(\beta \right))\text{.}$ Starting from $A\text{,}$ proceed by any rectangular route one square at a time until $B$ is reached. Counting $+1$ for each step made upwards or to the right and $-1$ for each step made downwards or to the left, the resultant number of steps made is the axial distance from $A$ to $B$. For squares belonging to distinct diagrams, axial distance is the distance of any rectangular route, counted as above, in the diagram obtained by superimposing $D\left(\beta \right)$ upon $D\left(\alpha \right)\text{.}$

Finally

Definition (1.1.16) The axial distance from the letter $p$ to the letter $q$ in a tableau $T^{\mu }$ is the axial distance from the square of $T^{\mu }$ in which $p$ appears to the square of $T^{\mu }$ in which $q$ appears.

Thus in Figure 2 the axial distance from $4$ to $13$ is $3,$ the axial distance from $13$ to $4$ is $-3,$ the axial distance from $6$ to $3$ is $-1$ and the axial distance from $9$ to $7$ is $0\text{.}$

The Semi-Normal Representations of the Symmetric Group

We briefly describe the irreducible semi-normal representations of the symmetric group $S_n$ on $n$ letters. The conjugacy classes of $S_n$ are parameterized by the partitions of $n\text{.}$ In ([You1930]) Young constructed for each partition $\left(\alpha \right)$ of $n$ an irreducible representation $\left[\alpha \right]$ of $S_n$ of degree $f^{\alpha }$ by constructing primitive idempotents, the natural idempotents, in the group algebra $\mathbb{Q}{S}_n$ from the standard tableaux of shape $\left(\alpha \right)\text{.}$ The distinctive feature of these representations is that they are integral, i.e., matrix representations afforded by the minimal left ideals generated by these idempotents have entries in $\mathbb{Z}\text{.}$ In a subsequent paper ([You1931]) Young constructed an equivalent form of these representations by means of the semi-normal idempotents. While the corresponding matrix representations are not integral, Young showed an elegant construction to hold for the matrices of the transpositions $(i-1,i)$ by means of the standard tableaux. For a tableau $T^{\alpha }$ of shape $\left(\alpha \right)\text{,}$ let $(i-1,i)T^{\alpha }$ denote the tableau obtained by interchanging the letters $i-1$ and $i$ in $T\text{.}$ If $T^{\alpha }$ is a standard tableau and the letters $i-1$ and $i$ do not occur either in the same row or column of $T^{\alpha }\text{,}$ then $(i-1,i)T^{\alpha }$ is again a standard tableau. Young's fundamental theorem giving the semi-normal form of the representations of $S_n$ can now be stated as follows.

Theorem (1.2.1) Let $T_1^{\alpha },\dots ,T_f^{\alpha },$ $f=f^{\alpha },$ be the arrangement of the standard tableaux of shape $\left(\alpha \right)$ according to the last letter sequence. To construct the $f\times f$ matrix representing $(i-1,i)$ in the irreducible representation $\left[\alpha \right]$ of $S_n$ corresponding to $\left(\alpha \right)\text{,}$ place

(i)	$1$ in the $p,p\text{-th}$ entry where $T_p^{\alpha }$ has $i-1$ and $i$ in the same row,
(ii)	$-1$ in the $p,p\text{-th}$ entry where $T_p^{\alpha }$ has $i-1$ and $i$ in the same column,
(iii)	the matrix $$\left(\begin{array}{cc}-\rho & \rho +1\\ \rho -1& \rho \end{array}\right)$$ in the $p,p\text{-th,}$ $p,q\text{-th,}$ $q,p\text{-th}$ and $q,q\text{-th}$ entries where $p<q,$ $T_q^{\alpha }=(i-1,i)T_p^{\alpha }$ and $1/\rho$ is the axial distance (see (1.1.6)) from $i$ to $i-1$ in $T_p^{\alpha }\text{,}$
(iv)	zeros elsewhere.

The importance of the semi-normal form is that it provides an inductive construction of the irreducible representations of $S_n$ and moreover is defined in terms of the generators and relations of $S_n\text{.}$

Young's fundamental theorem can be extended to yield representations of $S_n$ corresponding to double partitions $\left(\mu \right)=(\alpha ,\beta )$ of $n\text{.}$ If $\left(\alpha \right)$ is a partition of $k$ and $\left(\beta \right)$ is a partition of $l$ with $k+l=n\text{,}$ let $\left[\alpha \right]·\left[\beta \right]$ denote the representation of $S_n$ induced from the direct product representation $\left[\alpha \right]\times \left[\beta \right]$ of the subgroup $S_k\times S_l$ of $S_n\text{.}$ The representation $\left[\alpha \right]·\left[\beta \right]$ is called the outer product representation of $\left[\alpha \right]$ and $\left[\beta \right]\text{.}$ For a standard tableau $T^{\mu }=(T^{\alpha },T^{\beta })$ of shape $\left(\mu \right)=(\alpha ,\beta )\text{,}$ if the letters $i-1$ and $i$ do not occur either in the same row or column of $T^{\alpha }$ or $T^{\beta },$ $(i-1,i)T^u$ is again a standard tableau. Arranging the standard tableaux of shape $\left(\mu \right)$ according to the last letter sequence, we have (see [Rob1961], p. 54).

Theorem (1.2.2) To construct the matrices presenting $(i-1,i)$ in the outer product representation $\left[\alpha \right]·\left[\beta \right]$ of $S_n\text{,}$ apply the construction given in Theorem (1.2.1) to the standard tableau $(T^{\alpha },T^{\beta })$ of shape $(\alpha ,\beta ),$ setting $\rho =0$ in (iii) if the letters $i-1$ and $i$ belong to distinct tableaux $T^{\alpha }\text{,}$ $T^{\beta }\text{.}$

The hyperoctahedral group $H_n$ of order $2^{n}n!$ is the group of signed permutations on $n$ letters. It can be regarded as acting on an orthonormal basis $e_1,\dots ,e_n$ of ${ℝ}^n$ by means of permutations and sign changes. Denote the $k\text{-th}$ sign change, $e_k\mapsto -e_k$ by $-\left(k\right)\text{.}$ The set of transpositions $(i-1,i),$ $i=2,\dots ,n$ and the first sign change, $-\left(1\right),$ generate $H_n\text{.}$ In ([You1929]) Young showed the conjugacy classes of $H_n$ to be parameterized by double partitions $\left(\mu \right)=(\alpha ,\beta )$ of $n$ and constructed for each double partition $\left(\mu \right)$ an irreducible representation $\left[\mu \right]$ of $H_n$ of degree $f^{\mu }$ by constructing primitive idempotents in $\mathbb{Q}{H}_n$ analogous to the natural idempotent of $S_n\text{.}$ Young did not construct the analogous of the semi-normal idempotents for $H_n\text{.}$ It is implicit in his work, however, that a semi-normal form can be constructed for the representations $\left[\mu \right]$ using Theorem (1.2.2). In particular, using the matrices for the transpositions $(i-1,i)$ in the outer product representation $\left[\alpha \right]·[\beta ],$ we need only construct a matrix for the first sign change $-\left(1\right)$ which satisfies the relations of $H_n\text{.}$ It can readily be shown (see corollary (2.2.15)), using the inductive ordering provided by the last letter sequence on the standard tableaux of shape $(\alpha ,\beta ),$ that

Theorem (1.2.3) To construct the matrices representing $(i-1,i)$ in the irreducible representations $\left[\mu \right]=[\alpha ,\beta ]$ of $H_n\text{,}$ apply Theorem (1.2.2). To construct the matrix representing $-\left(1\right)$ place

1.	$1$ in the $p,p\text{-th}$ entry if the letter $1$ appears in the tableau $T_p^{\alpha }$ of $T_p^{\mu }=(T_p^{\alpha },T_p^{\beta }),$
2.	$-1$ in the $p,p\text{-th}$ entry if the letter $1$ appears in the tableau $T_p^{\beta }$ of $T_p^{\mu }=(T_p^{\alpha },T_p^{\beta }),$
3.	zeros elsewhere.

Thus, analogous to Theorem (1.2.1), the irreducible representations of $H_n$ can be defined inductively and in terms of generators and relations.

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