The orthogonality relations of multivariate Krawtchouk polynomials are discussed. In case of two variables, the necessary and sufficient conditions of orthogonality is given by Grünbaum and Rahman in [SIGMA 6 (2010), 090, 12 pages]. In this study, a simple proof of the necessary and sufficient condition of orthogonality is given for a general case.
In this paper, a factorization formula of Schur's Q-functions is shown. This formula gives us plethysms p r • Q λ. Moreover, some formulas of spin characters of symmetric groups are presented.
An expression is given for the plethysm p 2 • S , where p 2 is the power sum of degree two and S is the Schur function indexed by a rectangular partition. The formula can be well understood from the viewpoint of the basic representation of the affine Lie algebra of type A (2) 2
Formulas are obtained that express the Schur S-functions indexed by Young diagrams of rectangular shape as linear combinations of " mixed " products of Schur's Sand Q-functions. The proof is achieved by using representations of the affine Lie algebra of type A (1) 1. A realization of the basic representation that is of " D (2) 2 "-type plays the central… (More)
A new basis for the polynomial ring of infinitely many variables is constructed which consists of products of Schur functions and Q-functions. The transition matrix from the natural Schur function basis is investigated.
The symmetric group S 2n and the hyperoctaheadral group H n is a Gelfand triple for an arbitrary linear representation ϕ of H n. Their ϕ-spherical functions can be caught as transition matrix between suitable symmetric functions and the power sums. We generalize this triplet in the term of wreath product. It is shown that our triplet are always to be a… (More)