KOSTANT PARTITION FUNCTION FOR sl4(C) AND sp6(C)

ABSTRACT In this paper, we obtain a closed formula for the Kostant’s partition function for the Lie algebras sl4(C) and sp6(C): Using this function, one can compute the weight multiplicity of irreducible representations of the Lie algebras sl4(C) and sp6(C).


Introduction and preliminaries
One of the important problems in the representation theory of simple Lie algebras is the multiplicity computation of a weight in a finite dimensional complex irreducible representation of a Lie algebra. Let L be a finite dimensional complex semi-simple Lie algebra with a Cartan subalgebra H and root system Φ: Suppose λ is an integral dominant weight of L and VðλÞ is the corresponding irreducible L-module. For any other integral dominant weight μ; we denote the multiplicity of μ in λ by mðλ; μÞ: One way to compute mðλ; μÞ is by Kostant's weight multiplicity formula (Kostant, 1958): ðÀ 1Þ ,ðσÞ }ðσðλ þ ρÞ À ðμ þ ρÞÞ; (1) where W is the Weyl group, ,ðσÞ denotes the length of σ; ρ ¼ 1 2 P α2Φ þ α with Φ þ being the set of positive roots of L: In (1), } is the Kostant partition function. For any weight γ; }ðγÞ is the number of ways to write γ as a linear combination of positive roots with non-negative integral coefficients. In general, there is no known closed formula for the } on arbitrary weights of a Lie algebra. However, there has been some success in the low rank of some classical Lie algebras (Harris & Lauber, 2017;Refaghat & Shahryari, 2012) and for a particular weight (Harris et al., 2018). Moreover, many methods have been used to solve these types of problems (Adiga et al., 2016;Deckhart, 1985;Harris & Lauber, 2017;Kostant, 1958;Sarikaya et al., 2020;Srivastava & Chaudhary, 2015;Srivastava & Saikia, 2020;Zhang et al., 2009). In this paper, we are interested in finding a closed formula for } in Lie algebras sl 4 ðCÞ and sp 6 ðCÞ. To make different notations of Kostant's partition function in Lie algebras sl 4 ðCÞ and sp 6 ðCÞ, we use a notation } sl for Lie algebra sl 4 ðCÞ and } sp for Lie algebra sp 6 ðCÞ Theorem 1.1. Let γ ¼ aR 1 þ bR 2 þ cR 3 be a weight of the Lie algebra sl 4 ðCÞ where a; b and c are non-negative integers.

(i) If b � a and b � c; then
Theorem 1.2. Let γ ¼ aR 1 þ bR 2 þ cR 3 be a weight of the Lie algebra sp 6 ðCÞ, then

Background
In this section, we give a review of a set of notations of Lie algebras sl 4 ðCÞ and sp 6 ðCÞ A Cartan subalgebra for sl 4 ðCÞ is Also, the set is a basis for Φ: Then, the positive roots are the set We will denote the elements of Φ þ by β 1 ; . . . ; β 6 : If we write them as a linear combination of simple roots then we have The notations of sp 6 ðCÞ are similar. A Cartan subalgebra for sp 6 ðCÞ is H ¼ fh ¼ diagða 1 ; a 2 ; a 3 ; À a 1 ; À a 2 ; À a 3 Þja i 2 Cg; and the functional μ i is defined as above. A root system for sp 6 ðCÞ is the set and the simple roots are the set The positive roots of sp 6 ðCÞ are

Conclusion and suggestion
Kostant partition is a very important function in studying representations of Lie algebras. For instance, in Refaghat & Shahryari (2013), the Kostant partition function of Lie algebras of sp 4 ðCÞ is used to find representations of symmetry classes of tensors as sp 4 ðCÞmodule and thus, the branching rule A n ! C 2 is obtained. Now, this process can be repeated using the Kostant partition function of sp 6 ðCÞ to find branching rule A n ! C 3 or similar branching rules. Also, by developing this method, it can be tried to find the exact values of the Kostant partition function for Lie algebras slnðCÞ and sp 2n ðCÞ.

Disclosure statement
No potential conflict of interest was reported by the author(s).