On inequalities for normalized Schur functions

Abstract

We prove a conjecture of Cuttler et al. [2011] [A. Cuttler, C. Greene, and M. Skandera; Inequalities for symmetric means. European J. Combinatorics, 32(2011), 745-761] on the monotonicity of normalized Schur functions under the usual (dominance) partial-order on partitions. We believe that our proof technique may be helpful in obtaining similar inequalities for other symmetric functions.

Majorization and Spherical Functions

“…For any λ, µ ∈ W N , we have λ µ if and only if L λ ≥ L µ pointwise on R N . Theorem 1.1 generalizes an ex-conjecture of Cuttler, Greene, and Skandera [6] which characterizes the majorization order on Young diagrams via nonnegative specializations of Schur polynomials (the original conjecture was proved by Sra [33], and refined by Khare and Tao [20]). The purpose of this paper is to prove a general principle that yields a large family of majorization-characterizing inequalities in a very general context; in particular, Theorem 1.1 emerges as a special case of this construction.…”

On an integral representation of the normalized trace of the $k$-th symmetric tensor power of matrices and some applications

“…In [35], the author used the Harish-Chandra-Itzykson-Zuber (HCIZ) integral to represent the term-normalized Schur polynomials in order to obtain a monotonicity result concerning a conjecture formulated in [10]. As mentioned earlier, we shall next use the integral representation obtained in Section 2 to provide a generalization of the preceding theorem.…”

“…Our work is based on majorization methods for moments and thus differs from the algebraic approach used in [10]. Since the integrand in HCIZ involves two matrices, the author in [35] was able to express the term-normalized Schur polynomial as a single integral (over the set of unitary matrices). The existence of the exponential term in HCIZ was needed to obtain a Schur convexity result.…”

“…The case where the probability measure is considered on [0, ∞[ can be found in [27] (see p. 107). We formulate this result in the context of our work and we provide a proof motivated by the work of S. Sra in [35] which is based on a Schur's result for convex symmetric functions. Proof.…”

“…In connection with the analysis of symmetric polynomials, we recall that the Harish-Chandra-Itzykson-Zuber integral which was used by S. Sra in [35] to express the normalized Schur polynomials. Via such a representation together with a technique of Schur's convexity, the author in [35] presented the sufficiency for a conjecture that was proposed in [10] on the monotonicity (via a lexicographic order) for (a product of) normalized Schur polynomials. More explicitly, in [10] the authors presented many inequalities for symmetric functions and in particular, they proved the monotonicity for (a product of) normalized complete symmetric polynomials on R n + .…”