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Register a new userStaircase skew Schur functions are Schur P-positive
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We prove Stanleys conjecture that, if delta_n is the staircase shape, then the skew Schur functions s_{delta_n / mu} are non-negative sums of Schur P-functions. We prove that the coefficients in this sum count certain fillings of shifted shapes. In particular, for the skew Schur function s_{delta_n / delta_{n-2}}, we discuss connections with Eulerian numbers and alternating permutations.
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We provide a non-recursive, combinatorial classification of multiplicity-free skew Schur polynomials. These polynomials are $GL_n$, and $SL_n$, characters of the skew Schur modules. Our result extends work of H. Thomas--A. Yong, and C. Gutschwager, in which they classify the multiplicity-free skew Schur functions.
Cylindric skew Schur functions, a generalization of skew Schur functions, are closely related to the famous problem finding a combinatorial formula for the 3-point Gromov-Witten invariants of Grassmannian. In this paper, we prove cylindric Schur positivity of the cylindric skew Schur functions, conjectured by McNamara. We also show that all coefficients appearing in the expansion are the same as $3$-point Gromov-Witten invariants. We start discussing properties of affine Stanley symmetric functions for general affine permutations and $321$-avoiding affine permutations, and explain how these functions are related to cylindric skew Schur functions. We also provide an effective algorithm to compute the expansion of the cylindric skew Schur functions in terms of the cylindric Schur functions, and the expansion of affine Stanley symmetric functions in terms of affine Schur functions.
We give a basis for the space V spanned by the lowest degree part hat{s}_lambda of the expansion of the Schur symmetric functions s_lambda in terms of power sums, where we define the degree of the power sum p_i to be 1. In particular, the dimension of the subspace V_n spanned by those hat{s}_lambda for which lambda is a partition of n is equal to the number of partitions of n whose parts differ by at least 2. We also show that a symmetric function closely related to hat{s}_lambda has the same coefficients when expanded in terms of power sums or augmented monomial symmetric functions. Proofs are based on the theory of minimal border strip decompositions of Young diagrams.
We study the asymptotics of Schur polynomials with partitions $lambda$ which are almost staircase; more precisely, partitions that differ from $((m-1)(N-1),(m-1)(N-2),ldots,(m-1),0)$ by at most one component at the beginning as $Nrightarrow infty$, for a positive integer $mge 1$ independent of $N$. By applying either determinant formulas or integral representations for Schur functions, we show that $frac{1}{N}log frac{s_{lambda}(u_1,ldots,u_k, x_{k+1},ldots,x_N)}{s_{lambda}(x_1,ldots,x_N)}$ converges to a sum of $k$ single-variable holomorphic functions, each of which depends on the variable $u_i$ for $1leq ileq k$, when there are only finitely many distinct $x_i$s and each $u_i$ is in a neighborhood of $x_i$, as $Nrightarrowinfty$. The results are related to the law of large numbers and central limit theorem for the dimer configurations on contracting square-hexagon lattices with certain boundary conditions.
We give a new characterization of Littlewood-Richardson-Stembridge tableaux for Schur $P$-functions by using the theory of $mf{q}(n)$-crystals. We also give alternate proofs of the Schur $P$-expansion of a skew Schur function due to Ardila and Serrano, and the Schur expansion of a Schur $P$-function due to Stembridge using the associated crystal structures.
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