Some results on weight multiplicity for Lie groups

2014

We classify irreducible representations of the special linear groups in positive characteristic with small weight multiplicities with respect to the group rank and give estimates for the maximal weight multiplicities. For the natural embeddings of the classical groups, inductive systems of representations with totally bounded weight multiplicities are classified. An analogue of the Steinberg tensor product theorem for arbitrary indecomposable inductive systems for such embeddings is proved.

2021

Kostant's weight q-multiplicity formula is an alternating sum over a finite group known as the Weyl group, whose terms involve the q-analog of Kostant's partition function. The q-analog of the partition function is a polynomial-valued function defined by ℘_q(ξ)=∑_i=0^k c_i q^i, where c_i is the number of ways the weight ξ can be written as a sum of exactly i positive roots of a Lie algebra 𝔤. The evaluation of the q-multiplicity formula at q = 1 recovers the multiplicity of a weight in an irreducible highest weight representation of 𝔤. In this paper, we specialize to the Lie algebra 𝔰𝔭_6(ℂ) and we provide a closed formula for the q-analog of Kostant's partition function, which extends recent results of Shahi, Refaghat, and Marefat. We also describe the supporting sets of the multiplicity formula (known as the Weyl alternation sets of 𝔰𝔭_6(ℂ)), and use these results to provide a closed formula for the q-multiplicity for any pair of dominant integral weights of 𝔰𝔭_6(ℂ). Th...

Journal of Lie theory

The aim of this paper is to present a new character formula for finite-dimensional representations of finite-dimensional complex semisimple Lie Algebras and compact semisimple Lie Groups. Some applications of the new formula include the exact determination of the number of weights in a representation, new recursion formulas for multiplicities and, in some cases, closed formulas for the multiplicities themselves.

BIT, 1969

It is the purpose of this paper to provide an account of some experience that we have gained over the past year with machine computations of characters of irreducible representations of simple Lie algebras. These computations were for the most part carried out in the summer of 1968 on the UNIVAC 1108 computer at the Carnegie-Mellon University, using UNIVAC 1107/8 ALGOL and FORTRAN V implementations of the Dynkin and Freudenthal algorithms. An ALGOL 60 adaptation of these programs is given in the appendix to this paper. We have found from the actua] runs that these algorithms provide an entirely practical procedure for computing weight diagrams of Lie modules. In a survey of weight diagrams of irreducible modules of dimension up to 1000 for simple Lie algebras of rank up to 8, it was found that the FORTRAN version takes on the average only 2.5 seconds to compute a single weight diagram including multiplicities. The method is practical therefore for the special unitary groups up to SU(9), for the orthogonal groups up to S0(17), the symplectic groups up to Sp(8), as well as for all five exceptional simple Lie groups Gg, Fa, E6, E~ and Es, the last three of these having comparatively few representations of dimension less than a thousand however. These limits on rank and dimension are not rigid

arXiv (Cornell University), 2023

In the 1980s, Enright, Howe and Wallach [EHW] and independently Jakobsen [J] gave a complete classification of the unitary highest weight modules. In this paper we give a more direct and elementary proof of the same result for the (universal covers of the) Lie groups Sp(2n, R), SO * (2n) and SU (p, q). We also show how to describe the set of unitary highest weight modules with a given infinitesimal character.

2004

λ n fundamental weights, W the Weyl group of Φ, Z[Λ] the group ring of Λ over Z and Z[Λ] W the set of elements in Z[Λ] which are invariant under W. For a weight µ, we define the elementary symmetric sum S(e µ) of µ, the elementary alternating sum A(e µ) of µ and the character χ µ of µ as follows: S(e µ) = β∈Wµ e β , A(e µ) = w∈W det(w)e w(µ) and χ µ = A(e µ+δ) A(e δ) respectively, where δ is the half sum of all positive roots. Let S = S(e λ i) : 1 ≤ i ≤ n and χ = χ λ i : 1 ≤ i ≤ n be the set of elementary symmetric sums of fundamental weights and set of characters of fundamental weights, respectively. It is well-known that both S and χ are bases for Z-module Z[Λ] W. In this research, we are interested in finding relations between elements in the sets S and χ in the case of root systems whose Dynkin diagrams are A n , B n , C n , D n and G 2 for appropriate integers n.

Journal of Mathematical Sciences, 2009

Lower estimates for the maximal weight multiplicities in irreducible representations of the algebraic groups of types Bn, Cn, and Dn in positive characteristic p are found under some minor restrictions on p. If G = Bn(K), Cn(K), or Dn(K), n ≥ 8, p > 2 for types Bn and Dn and p > 7 for type Cn, then either the maximal weight multiplicity for an irreducible representation of G is at least n − 7 or all its weight multiplicities are equal to 1.

Journal of Mathematical Physics, 1998

A complementary group to SU(n) is found that realizes all features of the Littlewood rule for Kronecker products of SU(n) representations. This is accomplished by considering a state of SU(n) to be a special Gel'fand state of the complementary group U(2n − 2). The labels of U(2n − 2) can be used as the outer multiplicity labels needed to distinguish multiple occurrences of irreducible representations (irreps) in the SU(n)×SU(n) ↓ SU(n) decomposition that is obtained from the Littlewood rule. Furthermore, this realization can be used to determine SU(n) ⊃ SU(n−1)×U(1) Reduced Wigner Coefficients (RWCs) and Clebsch-Gordan Coefficients (CGCs) of SU(n), using algebraic or numeric methods, in either the canonical or a noncanonical basis. The method is recursive in that it uses simpler RWCs or CGCs with one symmetric irrep in conjunction with standard recoupling procedures. New explicit formulae for the multiplicity for SU(3) and SU(4) are used to illustrate the theory.

A matrix is said to be monomial if every row and column has only one non-zero entry. Let G be a group. A representation \rho: G \rightarrow GL_n(C) is said to be a monomial representation of G if there exists a basis with respect to which \rho(g) is a monomial matrixfor every g\in G. We use elementary methods to classify the irreducible monomial representations of the groups L_2(q), L_3(q) and their natural decorations.

Journal of Mathematical Sciences, 2010

Lower estimates for the maximal weight multiplicities in irreducible representations of the algebraic groups of type Cn in characteristic p ≤ 7 are found. If n ≥ 8 and p = 2, then for an irreducible representation either such a multiplicity is at least n − 4 − [n]4, where [n]4 is the residue of n modulo 4, or all the weight multiplicities are equal to 1. For p = 2, the situation is more complicated, and for every n and l there exists a class of representations with the maximal weight multiplicity equal to 2 l. For symplectic groups in characteristic p > 7 and spinor groups similar results were obtained earlier. Bibliography: 15 titles.