Cylindric symmetric functions and positivity

Christian Korff,David Palazzo

doi:10.5802/alco.90

Abstract

We introduce new families of cylindric symmetric functions as subcoalgebras in the ring of symmetric functions Λ (viewed as a Hopf algebra) which have non-negative structure constants. Combinatorially these cylindric symmetric functions are defined as weighted sums over cylindric reverse plane partitions or - alternatively - in terms of sets of affine permutations. We relate their combinatorial definition to an algebraic construction in terms of the principal Heisenberg subalgebra of the affine Lie algebra slˆn and a specialised cyclotomic Hecke algebra. Using Schur–Weyl duality we show that the new cylindric symmetric functions arise as matrix elements of Lie algebra elements in the subspace of symmetric tensors of a particular level-0 module which can be identified with the small quantum cohomology ring of the k-fold product of projective space. The analogous construction in the subspace of alternating tensors gives the known set of cylindric Schur functions which are related to the small quantum cohomology ring of Grassmannians. We prove that cylindric Schur functions form a subcoalgebra in Λ whose structure constants are the 3-point genus 0 Gromov–Witten invariants. We show that the new families of cylindric functions obtained from the subspace of symmetric tensors also share the structure constants of a symmetric Frobenius algebra, which we define in terms of tensor multiplicities of the generalised symmetric group G(n,1,k).

Highlights

The ring of symmetric functions Λ = ←lim− Λk with Λk = C[x1, . . . , xk]Sk lies in the intersection of representation theory, algebraic combinatorics and geometry
In order to motivate our results and set the scene for our discussion, we briefly recall a classic result for the cohomology of Grassmannians, which showcases the interplay between the mentioned areas based on symmetric functions
In this article we investigate the combinatorial structure of the q-deformed Verlinde algebra from [30] in the limit q → 1: we construct two families of positive subcoalgebras of Λ whose structure constants are given by Nλνμ(1); see Corollary 5.25

Summary

Introduction

We show equality between the images of Z(Hk) and the principal Heisenberg subalgebra in EndR Vk. Restricting to the subspace Vk− ⊂ Vk of alternating tensors, we recover the quantum cohomology of Grassmannians qH∗(Gr(k, n)) via the Satake correspondence: the Schur polynomials sλ(x1, . We show that the subspace spanned by these non-skew cylindric complete symmetric functions is a positive subcoalgebra of Λ, whose non-negative integer structure constants Nμλν coincide with those of the Frobenius algebra Vk+ and which we express in terms of tensor multiplicities of the generalised symmetric group G(n, 1, k). Cylindric elementary symmetric functions eλ/d/μ enter naturally by considering the image of the hλ/d/μ under the antipode which is part of the Hopf algebra structure on Λ Their combinatorial definition involves row strict cylindric reverse plane partitions; see Figure 1. The new aspect in our work is that we link this combinatorial realisation for the extended affine symmetric group Sk to cylindric loops and reverse plane partitions by considering the level-n action for hλ/d/μ and eλ/d/μ, while for the cylindric Schur functions we consider the shifted level-n action

The principal Heisenberg subalgebra

The affine symmetric group and wreath products

Frobenius structures on symmetric and alternating tensors

Cylindric symmetric functions