The symmetric functions catalog

An overview of symmetric functions and related topics

2019-05-20

Zonal symmetric functions

The zonal symmetric functions were introduced by Hua, in [Hua63]. For an introduction, see [Chapter 7, Mac95].

The zonal symmetric functions $\zonal_\lambda(\xvec)$ associated with the Gelfand pair $(\symS_{2n},H_n)$ are given by the specialization $\alpha=2$ in the Jack symmetric functions. That is, $\zonal_\lambda(\xvec) = \jackJ_\lambda(\xvec;2).$

Here, $H_n$ is the hyperoctahedral group of degree $n,$ given by the centralizer of the simple transpositions $(12),(34),\dotsc,(2n-1,2n).$ We have that $|H_n| = 2^n n!.$

Power sum expansion

In [FŚ11], the authors present a formula for $\zonal_\lambda(\xvec)$ as a signed sum over $T$-admissible pair-partitions.

$\zonal_\lambda(\xvec) = \sum_{(S_1,S_2) \text{ T-admissible}} (-1)^{L(S_1,S_2)} \powerSum_{L(S_1,S_2)}(\xvec).$

Here, $T$ is the standard Young tableau of shape $2\lambda,$ with $1,2,\dotsc,2\lambda_1$ in the first row, and so on. The expression $L(S_1,S_2)$ is the sizes of the components of certain bipartite graphs, where every vertex has degree $2,$ there are $2n$ edges labeled $1,\dotsc,2n$ and $\{i,j\} \in S_c$ if and only if edges $i$ and $j$ are share a vertex with color $c \in \{1,2\}.$

Pair-partitions can be interpreted as a disjoint product of transpositions, and thus as elements in $\symS_{2n}.$ Let $S = \{\{1,2\},\{3,4\},\dotsc,\{2n-1,2n\}\}.$ Let $T$ be a standard Young tableau. The pair $(S_1,S_2)$ is $T$-admissible if $S \circ S_1$ preserves the rows of $T,$ and $S_2$ preserves the columns.

References

• [FŚ11] Valentin Féray and Piotr Śniady. Zonal polynomials via Stanley's coordinates and free cumulants. Journal of Algebra, 334(1):338–373, May 2011.
• [Hua63] L. K. Hua. Harmonic analysis of functions of several complex variables in the classical domains (translations of mathematical monographs). American Mathematical Society, 1963.
• [Mac95] Ian G. Macdonald. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Second edition, 1995. With contributions by A. Zelevinsky, Oxford Science Publications