The Zelevinsky involution, co-standard modules for quantum affine algebras and interval symmetric functions

Abstract

Let $\mathscr{F}_n$ be the subcategory of finite-dimensional representations of the quantum loop algebra of $\mathfrak{sl}_{n+1}$ whose simple modules are indexed by multisegments consisting of intervals $[a,b]$, where $a,b$ are integers and $0<b-a\le n$. We define the co-standard module associated to a multisegment $\mathbf{s}$ to be the tensor product of the Kirillov--Reshetikhin modules, associated to the first node of the Dynkin diagram of $\mathfrak{sl}_{n+1}$, given by the segments of $\mathbf{s}$. Under a suitable (and necessary) restriction on $n$, we show that its socle is isomorphic to the simple module given by the Zelevinsky dual $\mathbf{s}^\dagger$ of $\mathbf{s}$. We then define an involution $Φ$ of the Grothendieck ring $\mathcal{K}(\mathscr {F}_n)$ and prove that it maps the class of a simple module to the class of the simple module given by its Zelevinsky dual. We also show that on suitable subcategories $Φ$ is induced by an exact autoequivalence reversing tensor products, and we give a $t$-analogue of the multiplicity identity in the quantum Grothendieck ring. Motivated by the determinantal formula for alternating snake modules, we realize $\mathcal{K}(\mathscr {F}_n)$ explicitly inside a polynomial ring in variables indexed by intervals, and, in the spirit of Macdonald&#39;s ninth variation of Schur functions, we show that the fundamental modules correspond to analogues of the elementary symmetric functions and the Kirillov--Reshetikhin modules to analogues of the complete homogeneous symmetric functions. This is compatible with $Φ$ and the classical involution $ω$, and we prove that alternating snake modules give a family of Schur positive symmetric functions extending the skew Schur functions.