On inequalities for normalized Schur functions

We prove a conjecture of Cuttler et al. (2011) on the monotonicity of normalized Schur functions under the usual (dominance) partialorder on partitions. We believe that our proof technique may be helpful in obtaining similar inequalities for other symmetric functions. © 2015 Elsevier Ltd. All rights reserved. We prove a conjecture of Cuttler et al. [1] on themonotonicity of normalized Schur functions under the majorization (dominance) partial-order on integer partitions. Schur functions are one of the most important bases for the algebra of symmetric functions. Let x = (x1, . . . , xn) be a tuple of n real variables. Schur functions of x are indexed by integer partitions = ( 1, . . . , n), where 1 · · · n, and can be written as the following ratio of determinants [7, pg. 49], [5, (3.1)]: s (x) = s (x1, . . . , xn) := det([x j+n j i ]i,j=1) det([x j i ]i,j=1) . (0.1) To each Schur function s (x) we can associate the normalized Schur function S (x) ⌘ S (x1, . . . , xn) := s (x1, . . . , xn) s (1, . . . , 1) = s (x) s (1n) . (0.2) Let , μ 2 Rn be decreasingly ordered. We say is majorized by μ, denoted μ, if k X i=1 i = k X i=1 μi for 1  i  n 1, and n X i=1 i = n X i=1 μi. (0.3) E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ejc.2015.07.005 0195-6698/© 2015 Elsevier Ltd. All rights reserved. S. Sra / European Journal of Combinatorics 51 (2016) 492–494 493 Cuttler et al. [1] studied normalized Schur functions (0.2) among other symmetric functions, and derived inequalities for them under the partial-order (0.3). They also conjectured related inequalities, of which perhaps Conjecture 1 is the most important. Conjecture 1 ([1]). Let and μ be partitions; and let x 0. Then, S (x)  Sμ(x), if and only if μ. Cuttler et al. [1] established necessity (i.e., S  Sμ only if μ), but sufficiency was left open. We prove sufficiency in this paper. Theorem 2. Let and μ be partitions such that μ, and let x 0. Then, S (x)  Sμ(x). Our proof technique differs completely from [1]: instead of taking a direct algebraic approach, we invoke a well-known integral from random matrix theory. We believe that our approach might extend to yield inequalities for other symmetric polynomials such as Jack polynomials [4] or even Hall–Littlewood and Macdonald polynomials [5]. 1. Majorization inequality for Schur polynomials Ourmain idea is to represent normalized Schur polynomials (0.2) using an integral compatiblewith the partial-order ‘ ’. One such integral is the Harish-Chandra–Itzykson–Zuber (HCIZ) integral [2,3]: I(A, B) := Z U(n) etr(U AUB)dU = cn det([eaibj ]i,j=1) (a) (b) , (1.1) where dU is the Haar probability measure on the unitary group U(n); a and b are vectors of eigenvalues of the Hermitian matrices A and B; is the Vandermonde determinant (a) := 1i 0 (for xi = 0, apply the usual continuity argument). Then, there exist reals a1, . . . , an such that eai = xi, whereby S (x1, . . . , xn) = s (elog x1 , . . . , elog xn) s (1, . . . , 1) = I(log X, B( )) E(log X) , (1.6) 494 S. Sra / European Journal of Combinatorics 51 (2016) 492–494 where X = Diag([xi]i=1); we write B( ) to explicitly indicate B’s dependence on as in Proposition 3. Since E(log X) > 0, to prove Theorem 2, it suffices to prove Theorem 4 instead. Theorem 4. Let X be an arbitrary Hermitian matrix. Define the map F : Rn ! R by F( ) := I(X,Diag( )), 2 Rn. Then, F is Schur-convex, i.e., if , μ 2 Rn such that μ, then F( )  F(μ). Proof. We know from [6, Proposition C.2, pg. 97] that a convex and symmetric function is Schurconvex. From the HCIZ integral (1.1) symmetry of F is apparent; to establish its convexity it suffices to demonstrate midpoint convexity: F +μ 2  2F( ) + 2F(μ) for , μ 2 Rn. (1.7) The elementary manipulations below show that inequality (1.7) holds. F +μ 2 = Z U(n) exp tr ⇥ U⇤XU Diag +μ 2 ⇤ dU