The adjoint representation of a classical Lie algebra and the support of Kostant’s weight multiplicity formula

Abstract

Even though weight multiplicity formulas, such as Kostant's formula, exist their computational use is extremely cumbersome. In fact, even in cases when the multiplicity is well understood, the number of terms considered in Kostant's formula is factorial in the rank of the Lie algebra and the value of the partition function is unknown. In this paper we address the difficult question: What are the contributing terms to the multiplicity of the zero weight in the adjoint representation of a finite dimensional Lie algebra? We describe and enumerate the cardinalities of these sets (through linear homogeneous recurrence relations with constant coefficients) for the classical Lie algebras of Type B, C, and D, the Type A case was computed by the first author in [5]. In addition, we compute the cardinality of the set of contributing terms for non-zero weight spaces in the adjoint representation. In the Type B case, the cardinality of one such non-zero-weight is enumerated by the Fibonacci numbers. We end with a computational proof of a result of Kostant regarding the exponents of the respective Lie algebra for some low rank examples and provide a section with open problems in this area.

Computing weight $q$-multiplicities for the representations of the simple Lie algebras

“…Notice that this element will also have simple reflections appended to it as i increases and eventually reaches this element. This process is guaranteed to work because we know that once the expression σ(λ + ρ) − (µ + ρ) yields a negative coefficient when expressed as a sum of simple roots, appending more reflections to σ which make the length of σ increase will only decrease the coefficients of the simple roots further, as was proven by the first and second author in [18,Proposition 3.4]. We present this result below, but omit its proof as it is both technical and lengthy.…”

“…For example, Harris showed that the number of Weyl group elements contributing nontrivially to the multiplicity of the zero weight in the adjoint representation (the representation with highest weight equal to the highest root) of slr+1(C) was given by Fibonacci numbers [16]. Later, Harris, Insko, and Williams, generalized these results to show that the number of Weyl group elements contributing nontrivially to the multiplicity of the zero weight in the adjoint representation of all classical Lie algebras is governed by linear homogeneous recurrence relations with constants coefficients [18].…”

“…5 Alternation set for E 8 +36q 4 +340q 5 +1719q 6 +5615q 7 + 13180q 8 + 23550q 9 + 33523q 10 + 39143q 11 + 38496q 12 + 32431q13 + 23801q 14 + 15354q 15 + 8802q16 + 4493q 17 +2057q 18 +839q 19 +306q 20 + 97q21 + 27q 22 + 6q23 + q 24 + 102q 4 + 811q 5 + 3664q 6 + 11081q 7 + 24516q 8 + 41972q 9 + 57833q 10 + 66092q 11 + 64180q 12 + 54006q13 + 39999q 14 + 26401q15 + 15677q16 + 8441q 17 + 4138q18 + 1851q19 + 753q 20 + 278q 21 + 91q22 + 26q23 + 6q24 + q 25 + 97q 4 + 800q 5 + 3701q 6 + 11450q 7 + 25774q 8 + 44828q 9 + 62575q 10 + 72345q 11 + 70856q 12 + 60013q13 + 44607q 14 + 29503q 15 + 17506q16 + 9405q 17 + 4585q18 + 2038q19 + 820q 20 + 299q 21 + 96q22 + 27q23 + 6q24 + q 25 +33q 4 +328q 5 +1717q 6 +5695q 7 + 13380q 8 + 23802q 9 + 33596q 10 + 38910q 11 + 37945q 12 + 31773q 13 + 23190q 14 + 14913q 15 + 8525q16 + 4356q 17 +1997q 18 +820q 19 +301q 20 + 97q21 + 27q 22 + 6q23 + q 24 +39q 4 +359q 5 +1795q 6 +5832q 7 + 13587q 8 + 24123q 9 + 34007q 10 + 39267q 11 + 38051q 12 + 31585q 13 + 22834q 14 + 14563q 15 + 8280q16 + 4226q 17 +1944q 18 +804q 19 +298q 20 + 97q21 + 27q 22 + 6q 23 + q 24 + 101q 4 + 801q 5 + 3610q 6 + 10895q 7 + 24090q 8 + 41252q 9 + 56831q 10 + 64856q 11 + 62802q 12 + 52626q13 + 38783q 14 + 25476q 15 + 15076q16 + 8103q 17 + 3980q18 + 1790q19 + 735q 20 + 274q 21 + 91q22 + 26q23 + 6q24 + q 25 + 98q 4 + 795q 5 + 3657q + 11248q 7 + 25275q 8 + 43902q + 61256q 10 + 70674q 11 + 69025q 12 + 58190q13 + 43057q 14 + 28346q 15 + 16787q16 + 9016q 17 + 4413q + 1973q19 + 802q 20 + 295q 21 + 96q…”

Pamela Harris: The Mathematical Rise and Social Contribution of a Dreamer

“…This formula is an impractically large sum over the elements of the corresponding Weyl group. For the classical Lie algebras, Pamela [3] and her collaborators [4] determined that the vast majority of the terms in this sum vanish, and they enumerated the contributing terms; these are rather unexpected results on a central object in mathematics.…”

Weight q-multiplicities for Representations of sp4(C)

“…This is due to the fact that in general the number of terms appearing in the sum are factorial in the rank of the Lie algebra and there is no known closed formula for the partition function involved. There has been recent progress in addressing these complications for particular weight multiplicity computations [4,5,7,8].…”