Zonal polynomials and domino tableaux

✍ Scribed by Nantel Bergeron; Adriano M. Garsia

• Book ID: 103060164
• Publisher: Elsevier Science
• Year: 1992
• Tongue: English
• Weight: 632 KB
• Volume: 99
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(92)90360-r

✦ Synopsis

Bergeron,

N. and A.M. Garsia, Zonal polynomials and domino tableaux, Discrete Mathematics 99 (1992) 3-15.

Let H be a subgroup of a finite group G. Define the element 0 of the group algebra d(G) by @= C&H h/lH(. This element is an idempotent which may be used to project from -Pa(G) to the linear span of the left cosets of H in G. If (H, G) is a Gelfand pair then the decomposition of 8 into minimal idempotents yields a useful basis for the Hecke algebra X(H, G). When this decomposition is applied to the pair (B,, S,) the resulting minimal idempotents are intimately related to the zonal polynomials.

In fact, the latter are the images of the minimal idempotents under an analogue of the Frobenius map. We show here that the Fourier transform of the minimal idempotents is supported by standard domino tableaux. We also give a multiplication algorithm for the zonal polynomials and relate the expansion coefficients to the Littlewood-Richardson's coefficients.