Interaction of Combinatorics and Representation Theory

150 Pages·2001·14.349 MB·English

by  John R. Stembridge, Jean-Yves Thibon, Marc A.A. van Leeuwen

Interaction of Combinatorics
nd
Representation Theory
Authors
Jean-Yves Thibon Marc A. A. van Leeuwen
$DepartmentofMatJohnRStembr_{B}ideematics$
Institut Gaspard Monge Universit\’e de Poitiers
University of Michigan Universit\’e de Marne-la-Vall\’ee D\’epartement de Math\’ematiques
Ann Arbor Cit\’e Descartes UFR Sciences $SP2MI$
Michigan 48109-110 5 Boulevard Descartes BPT\’el\’eport 2, 30179
U.S.A Champs-sur-Marne 86962 Futuroscope Chasseneuil Cedex
77454 Marne-la-Vall\’ee cedex France
France
AMS Subject Classifications: $05Exx,$ $05E05,05E10,05E15,17B10$
$17B37,20C30,20F55,68W30$
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Preface
This anthology of three papers is a fruitful product of the Research Project of
RIMS (Research Institute for Mathematical Sciences, Kyoto University) on “Com-
binatorial Methods in Representation Theory” for the academic year 1998-99.
The authors were all participants of the above project and played active roles
during that period. The following is a brief summary of the papers.
Prof. Stembridge’s paper gives a nice integrated survey on methods for actual
computation of basic representation-theoretic data for semi-simple Lie algebras
(over the complex numbers, say), such as weight multiplicities and tensor product
decompositions, and related structural data for their Weyl groups. Prof. Stem-
bridge has been engaged in developing packages of functions in MAPLE named
“Coxeter” and “Weyl” to deal with these problems, and this article reflects his
own experience in designing these programs. It also mentions connections with
recent research interests.
Prof. Thibon’s paper is a survey of the theory of the noncommutative sym-
metric functions initiated by the author and others (I.M. Gelfand, D. Krob, A.
Lascoux, B. Leclerc, V. Retakh et al.). It covers many topics in such areas as com-
binatorics, Lie algebra, the symmetric groups, the Hecke algebra and the quantum
group of type A and now the subject becomes one of the most active areas in
combinatorics.
Prof. Thibon shows that the framework of the noncommutative symmetric
functions leads us in a natural way to many noncommutative analogues and q-
analogues ofknown and famous results in each of these field and this article conveys
a good flavor of the theory.
Prof. van Leeuwen’s paper is a variation on the theme of the Littlewood-
Richardson rule, which plays an important role both in combinatorics and repre-
sentation theory of the classical groups. Prof. van Leeuwen presents a proof and
a unified perspective of the Littlewood-Richardson rule based on the “modern “
(post-1980) technology such as tableau switching, dual equivalence, and coplactic
operations. Also in this article he tries to give proofs of some results which are
hard to find in published literature, though they may be known to the experts. It
makes this article an introductory and self-contained exposition of the Littlewood-
Richardson rule and related combinatorial constructions.
$v$
All the articles are very pleasantly written and we hope the readers find in
them some excellent examples of outcomes of a happy marriage of combinatorics
and representation theory.
Masaki Kashiwara
Kazuhiko Koike
Soichi Okada
Itaru Terada
Hiro-Fumi Yamada
May 2001, Japan
vi
Table of Contents
Preface by Masaki Kashiwara, Kazuhiko Koike, Soichi Okada, Itaru Terada,
and Hiro-Fumi Yamada
Part 1 Computational Aspects of Root Systems, Coxeter
Groups and Weyl Characters 1
by John R. Stembridge
Part 2 Lectures on Noncommutative Symmetric Functions 39
by Jean-Yves Thibon
Part 3 The Littlewood-Richardson Rule, and Related
Combinatorics 95
by Marc A. A. van Leeuwen
vii
Computational Aspects ofRoot Systems,
Coxeter Groups, and Weyl Characters
John R. STEMBRIDGE
Table of Contents
0. Introduction 2
1. Reduced Words 2
2. Permutation Representations 4
3. The Conjugacy Problem 6
A. Characteristic polynomials 7
B. Permutation representations 7
C. Centrally symmetric orbits 8
D. Canonical representatives 9
4. Tkaversal 13
A. Finite automata . 14
B. A low-technology solution 15
C. Implementation . . . 17
D. Canonical descendants in Coxeter groups 18
5. The Weight System ofa Weyl Character 19
A. Weyl characters. . . . . . 20
B. The partial ordering of integral weights 21
C. The complexity of a representation 22
D. Generating the weight system 22
6. Weight Multiplicities 26
A. Freudenthal’s formula 26
B. The Moody-Patera refinement . . 27
C. The q-analogue of weight multiplicity 28
7. Tensor Product Multiplicities 29
A. The Brauer-Klimyk formula . 29
B. Double specialization of Weyl characters 31
C. The support of a tensor product 33
D. The qtensor algorithm 34
Partially supported by NSF Grant DMS-9700787 and RIMS, Kyoto University.
I would like to thank RIMS for their kind hospitality during the preparation of this article.
2 JOHN R. STEMBRIDGE
0. Introduction
In this article, our goal is to survey some of the fundamental computational
problems that arise in working with the structures mentioned in the title. We
became interested in these problems in the course of trying to gather data (and
prove theorems) involving the exceptional groups and their root systems, and this
in turn led us to the ongoing development of the Maple packages coxeter and weyl.
For the classical cases, especially type $A$, many of these problems are easy or
have well-known solutions. However these solutions often do not generalize. Here
our emphasis is on algorithms that are (for the most part) independent of the
classification of root systems.
The canonical example we always have in mind is .
$E_{8}$
We should remark that there are many researchers elsewhere who have also
developed software for these and similar problems; for example, there is the $LiE$
package of van Leeuwen, Cohen and Lisser (et. al.), the CHEVIE package for GAP
and Maple by Meinolf Geck (et. al.), and the Schur package of Brian Wybourne.
Web links to these packages can be found at the end of the article.
Throughout, will denote a finite crystallographic root system of rank em-
$\Phi$ $n$
bedded in a real Euclidean space $V$ with inner product $\{$ , $\}$. We let $\alpha_{1},$ $\ldots,$ $\alpha_{n}$
denote a collection of simple roots, with $\Phi^{+}$ the corresponding set of positive roots.
For any root $\alpha$, we write $\alpha^{\vee}$ $:=2\alpha/\langle\alpha,$ $\alpha$} for the corresponding co-root. We will
assume that the reader is familiar with the basic terminology of root systems and
reflection groups, as well as the classification of root systems by Dynkin diagrams.
Standard references are [Bo] and [H1-2].
The crystallographic hypothesis is unnecessary for much of what we discuss in
\S \S 1-4, however it introduces unpleasant computational details (e.g., the need for
floating-point or exact number field arithmetic) that would distract us from the
main issues. In \S4 we will temporarily relax the assumption of finiteness.
For nonzero $\alpha\inV$, we let $\sigma_{\alpha}\inGL(V)$ denote the corresponding reflection; i.e.,
. (0.1)
$\sigma_{\alpha}(\lambda)=\lambda-\langle\lambda,\alpha^{\vee}\rangle\alpha$ $(\lambda\inV)$
The Weyl group corresponding to is the (finite) group generated by the
$M^{r}/$ $\Phi$
reflections corresponding to the simple roots. For brevity. let .
$\sigma_{\alpha_{1}},$ $\ldots,$ $\sigma_{\alpha_{n}}$ $s_{i}=\sigma_{\alpha:}$
It is well-known that $W$ is a Coxeter group; i.e., the relations
$(s_{i}s_{j})^{m(i,j)}=1$, (0.2)
where $m(i,j)$ denotes the order of in $W$, define a presentation of $W$.
$s_{i}s_{j}$
1. Reduced Words
In most cases, the preferred data structure we use for representing Weyl group
elements are words (integer lists) that encode products of simple reflections. (An
alternative is discussed in the following section.) Thus the word encodes
$(i_{1},\ldots,i_{l})$
COMPUTATIONAL ASPECTS OF ROOT SYSTEMS 3
the group element . In these terms, group multiplication is concate-
$w=s_{i_{1}}\cdotss_{i_{l}}$
nation, and group inversion is reversal. Of course all that this does is to move the
real problem elsewhere. For this data structure, the problem is to decide when two
words encode the same group element, or to produce a canonical (minimum-length)
representative of a given group element.
While it possible to solve these word problems using only the Coxeter rela-
tions (0.2) (or the braid relations), there are much faster and simpler solutions
available that take advantage of the geometrical tools provided by the root system.
It is well-known that the hyperplanes
$\alpha^{\perp}=\{\lambda\inV:\{\lambda,\alpha\}=0\}$ $(\alpha\in\Phi)$
are stable under the action of $W$ and their removal from $V$ partitions the remainder
into connected components (chambers).
The action of $W$ on chambers is simply transitive. Thus if $\lambda\inV$ is any vector
in general position (i.e., not orthogonal to any root), then the words
$(i_{1},\ldots,i_{l})$
and $(j_{1},\ldots,j_{m})$ encode the same group element if and only if
.
$s_{i_{1}}\cdotss_{i},$ $(\lambda)=s_{j_{1}}\cdotss_{j_{m}}(\lambda)$
The cost of such acomputation amounts to $l+m$ vector additions, scalar mul-
tiplications, and scalar products (cf. (0.1)). However, we should point out that
the real cost is usually far less than would be incurred if the vectors involved were
randomly distributed. Indeed, in the standard realization of every crystallographic
root system, many of the roots (in some cases, all) have only one or two nonzero
coordinates relative to some orthonormal basis. If the code for performing vector
operations is written to take advantage of this sparsity, then the real cost of a vec-
tor operation involving a root is (often) the same as the cost of one or two scalar
operations.
The minimum length among all expressions for $w\in W$ is denoted $\ell(w)$.
To determine a canonical representation for the group element indexed by the
word , one may make use of the fact (e.g., see [H2, \S 5.4]) that
$(i_{1},\ldots,i_{l})$
$\ell(s_{i}w)<\ell(w)\Leftrightarroww^{-1}\alpha_{i}\in-\Phi^{+}$.
Indeed, it follows that if is any point in the fundamental chamber; i.e.,
$\lambda$
$\{\lambda,\alpha_{i}\}>0$ $(1\leq i\leq n)$,
then
$\ell(s_{i}w)<\ell(w)\Leftrightarrow\{w\lambda,\alpha_{i}\}<0$. (1.1)
In other words, has a minimum-length expression that begins with if and only
$w$ $s_{i}$
if $\{w\lambda,\alpha_{i}\}<0$. Therefore, we can determine the lexicographically first minimal
expression for $w$ by first computing $\mu$ $:=w\lambda$ (using any representation of $w$ as a
product of simple reflections), and then starting with the empty word,
1. Find the least index such that .
$i$ $\langle\mu,\alpha_{i}\rangle<0$
2. Append to the word being constructed.
$i$
3. Replace , and repeat.
$\mu\leftarrows_{i}\mu$
The algorithm terminates when reaches the fundamental chamber.
$\mu$
4 JOHN R. STEMBRIDGE
2. Permutation Representations
In some cases, it is preferable to use permutation representations of Weyl groups,
rather than reduced words. For example, this allows one to take advantage of the
extensive library ofgroup-theoretic tools (available in GAP, for example) that have
been developed over many years by the computational group theory community.
A basic issue that arises is the problem of converting between the two ways of
representing group elements. One direction is trivial. If we have permutations
representing the action of the simple reflections, it is easy to deter-
$\pi_{1},$ $\ldots,$ $\pi_{n}$
mine the permutation that corresponds to the group element encoded by the word
. The inverse problem is more significant.
$(i_{1},\ldots,i_{l})$
PROBLEM 2.1. Given permutation ofsome finite set, find (ifpos-
$s\sigma,$ $\pi_{1},$ $\ldots,$ $\pi_{7l}$
sible) an expression for of the form
$\sigma$
$\sigma=\pi_{i_{1}}^{\pm1}\cdots\pi_{i}^{\pm1}$
This is one of the fundamental problems of computational group theory1 and
fortunately there are good, polynomial-time algorithms for it that are based on
building a strong generating set in the sense of Sims [S].
For Weyl groups, one can use the geometry of root systems to quickly build a
strong generating set, much faster than is possible for general permutation groups.
First, we need to construct a permutation representation of the Weyl group $W$.
The natural way to do this is to let $W$ act on cosets of some subgroup. The most
convenient available subgroups are the so-called parabolic subgroups, the subgroups
generated by subsets of the simple reflections. Given any subset $J\subseteq\{1,\ldots,n\}$,
we let $W_{J}$ denote the parabolic subgroup of $W$ generated by $\{s_{i}: j\inJ\}$.
The coset space $W/W_{J}$ has a geometric representation as the orbit of a suitably
chosen point $\lambda\inV$. Indeed, the stabilizer subgroup ofevery point in $V$ is generated
by reflections, and the stabilizer of every point in the closure of the fundamental
chamber is generated by simple reflections (e.g., [H2, \S 1.12]). Thus by selecting
$\lambda$
so that
$\langle\lambda,\alpha_{i}\rangle=0$ $(i\in J)$; $(\lambda,\alpha_{i}\rangle$ $>0 (i\not\in J)$, (2.1)
we obtain a vector whose stabilizer is $W_{J}$, and the map $W/W_{J}\rightarrowW\lambda$ given by
$wW_{J}\vdash\precw\lambda$
is an isomorphism of sets-with-W-action. It is now easy to determine the permu-
tations $\pi_{1},$ $\ldots,$ $\pi_{n}$ that represent the action of the simple reflections on $W/W_{J}$.
Simply construct the orbit of $\lambda$ by starting with $\mathcal{O}=\{\lambda\}$ and then successively
add new nlembers to $\mathcal{O}$ until it is saturated under the action of . The
$s_{1},$ $\ldots,$ $s_{n}$
permutation can then be obtained by examining the action of on O.
$\pi_{i}$ $s_{i}$
We now have some permutation representations available, but are they faithful?
lIt is also the key to novelties such as Rubik’s cube.
COMPUTATIONAL ASPECTS OF ROOT SYSTEMS 5
$A_{n}$ $B_{n},C_{n}$ $D_{n}$ $E_{6}$ $E_{7}$ $E_{8}$ $F_{4}$ $G_{2}$
$n+1$ $2n$ $2n$ $27$ $56$ $240$ $24$ $6$
TABLE I. Degrees of permutation representations.
PROPOSITION 2.2. The permutation representation of $W$ on $W/W_{J}$ is faithful
if and only if $J$ omits at least on$e$ node from each connected component of the
diagram of$\Phi$. In particular, if$W$ is irreducible, the representation is faithful ifand
only if is a proper subgroup.
$W_{J}$
Proof. Since $W$ is the direct product of its irreducible components, it suffices
to restrict our attention to the irreducible case. Clearly, the hypothesis that
$W_{J}$
is a proper subgroup is necessary. Conversely, suppose $j\not\inJ$ and let $w\in W$ be
an element that acts trivially on $W/W_{J}$. Given $\lambda$ as described in (2.1), $w$ must
stabilize every vector $x\lambda$ with $x\in W$. Hence $w$ stabilizes $x\lambda-xs_{j}\lambda=\langle\lambda,\alpha_{j}^{\vee}\ranglex\alpha_{j}$.
However { $>0$, so stabilizes every ; i.e., fixes every point in the
$\lambda,$ $\alpha_{j}\rangle$ $w$ $x\alpha_{j}$ $w$
W-orbit of . Since the diagram of is assumed to be connected, this means that
$\alpha_{j}$ $\Phi$
$w$ stabilizes every simple root. (If $\alpha_{1}$ is in the span of $W\alpha_{j}$ and $\alpha_{2}$ is adjacent to
in the diagram, then for some nonzero scalar , so is also
$\alpha_{1}$ $s_{2}\alpha_{1}=\alpha_{1}+c\alpha_{2}$ $c$ $\alpha_{2}$
in the span of ) Hence acts as the identity map on V.
$W\alpha_{j}.$ $w$ $\square$
In Table I, we list the degrees of the smallest (faithful) parabolic permutation
representations in the irreducible cases.
Let us now turn to strong generating sets and the solution of Problem 2.1. For
a more comprehensive account, see [BLS] and the references cited there.
Let $S=\{\pi_{1},\ldots,\pi_{n}\}$ be a collection of permutations ofsome finite set $X$, and let
$G$ be the permutation group generated by $S$. A stabilizer chain for $G$ is a sequence
of subgroups
$G=G_{0}\supset G_{1}\supset\cdots\supset G_{l}=\{1\}$,
such that $G_{i}$ is the stabilizer in $G_{i-1}$ of some point $x_{i}\in X$. The set of points $B=$
$\{x_{1},\ldots,x_{l}\}$ is called a base. For $1\leq i\leq l$, let $S_{i}$ be a set of coset representatives
for , so that
$G_{i-1}/G_{i}$
.
$G_{i-1}=\bigcup_{\sigma\inS_{i}}\sigmaG_{i}$
Notice that each $\pi\in G$ has a unique representation of the form
, (2.2)
$\pi=\sigma_{1}\sigma_{2}\cdots\sigma_{l}$ $(\sigma_{i}\inS_{i})$
so in particular $S_{1}\cup\cdots\cupS_{l}$ generates $G$; it is called a strong generating set.
Once a strong generating set has been found, finding the factorization (2.2) is
rapid. Indeed, since $G_{i}$ is the stabilizer of $x_{i}$ in $G_{i-1}$ , there is a one-to-one corre-
spondence between $S_{i}$ and the $G_{i-1}$-orbit of $x_{i}$. We can even label the members of
by the points in this orbit so that the representative indexed by is the unique
$S_{i}$ $x$