Combinatorial Representation Theory

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

Last update: 17 September 2013

Part II

6. Generalizations of $GL(n,ℂ)$ results

There have been successful generalizations of the $GL(n,ℂ)$ results for the questions (Ia-c), (S1), (S2) to the following classes of groups and algebras.

(1)	Connected complex semisimple Lie groups.
	Examples: $SL(n,ℂ),$ $SO(n,ℂ),$ $Sp(2n,ℂ),$ $PGL(n,ℂ),$ $PSO(2n,ℂ),$ $PSp(2n,ℂ)\text{.}$
(2)	Compact connected real Lie groups.
	Examples: $SU(n,ℂ),$ $SO(n,ℝ),$ $Sp\left(n\right),$ where $Sp\left(n\right)=Sp(2n,ℂ)\cap U(2n,ℂ)\text{.}$
(3)	Finite dimensional complex semisimple Lie algebras.
	Examples: $𝔰𝔩(n,ℂ),$ $𝔰𝔬(n,ℂ),$ $𝔰𝔭(2n,ℂ)\text{.}$ See Appendix A7 for the complete list of the finite dimensional complex semisimple Lie algebras.
(4)	Quantum groups corresponding to complex semisimple Lie algebras.

The method of generalizing the $GL(n,ℂ)$ results to the objects in (1-4) is to reduce them all to case (3) and then solve case (3). The results for case (3) are given in Section 4. The reduction of cases (1) and (2) to case (3) are outlined in [Ser1987], and given in more detail in [Var1984] and [BtD1985]. The reduction of (4) to (3) is given in [CPr1994] and in [Jan1995].

Partial results for further generalizations

Some partial results along the lines of the results (Ia-c) and (S1-2) for $GL(n,ℂ)$ and complex semisimple Lie algebras have been obtained for the following groups and algebras.

(1)	Kac-Moody Lie algebras and groups
(2)	Yangians
(3)	Simple Lie superalgebras

Other groups and algebras, for which the combinatorial representation theory is not understood very well, are

(4)	Finite Chevalley groups
(5)	$𝔭\text{-adic}$ Chevalley groups
(6)	Real reductive Lie groups
(7)	The Virasoro algebra

There are many many possible ways that we could extend this list but probably these four cases are the most fundamental cases where the combinatorial representation theory has not been formulated. There has been intense work on all of these cases, but hardly any by combinatorialists. Thus there are many beautiful results known but very few of them have been stated or interpreted through a combinatorialists eyes. The world is a gold mine, yet to be mined!

Notes and references

(1)	An introductory reference to Kac-Moody Lie algebras is [Kac1104219]. This book contains a good description of the basic representation theory of these algebras. We don’t know of a good introductory reference for the Kac-Moody groups case, we would suggest beginning with the paper [KKu0866159] and following the references there.
(2)	The basic introductory reference for Yangians and their basic representation theory is [CPr1994], Chapter 12. See also the references given there.
(3)	The best introductory reference for Lie superalgebras is Scheunert’s book [Sch1979]. For an update on the combinatorial representation theory of these cases see the papers [Ser1984], [BRe1987], [BSR1998], and [Ser1997].
(4)	Finding a general combinatorial representation theory for finite Chevalley groups has been elusive for many years. After the fundamental work of J.A. Green [Gre1955] in 1955 which established a combinatorial representation theory for $GL(n,{𝔽}_q)$ there has been a concerted effort to extend these results to other finite Chevalley groups. G. Lusztig [Lus1977,Lus1977-2,Lus1984,Lus1974] has made important contributions to this field; in particular, the results of Deligne-Lusztig [DLu1976] are fundamental. However, this is a geometric approach rather than a combinatorial one and there is much work to be done for combinatorialists, even in interpreting the known results from combinatorial viewpoint. A good introductory treatment of this theory is the book by Digne and Michel [DMi1991]. The original work of Green is treated in [Mac1995] Chapt. IV.
(5)	The representation theory of $𝔭\text{-adic}$ Lie groups has been studied intensely by representation theorists but essentially not at all by combinatorialists. It is clear that there is a beautiful (although possibly very difficult) combinatorial representation theory lurking here. The best introductory reference to this work is the paper of R. Howe [How1994] on $𝔭\text{-adic}$ $GL\left(n\right)\text{.}$ Recent results of G. Lusztig [Lus1995-2] are a very important step in providing a general combinatorial representation theory for $𝔭\text{-adic}$ groups.
(6)	The best place to read about the representation theory of real reductive groups is in the books of D. Vogan and N. Wallach [KVo1995], [Vog1987] , [Vog1981], [Wal1988,Wal1992].
(7)	The Virasoro algebra is a Lie algebra that seems to turn up in every back alley of representation theory. One can only surmise that it must have a beautiful combinatorial representation theory that is waiting to be clarified. A good place to read about the Virasoro algebra is in [FFu1990].

Notes and references

This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.

Key words and phrases. Algebraic combinatorics, representations.

Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..

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