Generalized F-signatures of Hibi rings

TL;DR

This paper introduces methods to compute the generalized F-signature of Hibi rings, a class of toric rings, by counting elements in symmetric groups, and applies these to Segre products of polynomial rings.

Contribution

It provides novel computational techniques for the generalized F-signature of Hibi rings, linking algebraic invariants to combinatorial group counting.

Findings

Computed generalized F-signatures for Hibi rings.

Established a counting method using symmetric group elements.

Derived formulas for F-signatures of Segre products of polynomial rings.

Abstract

The $F$-signature is a numerical invariant defined by the number of free direct summands in the Frobenius push-forward, and it measures singularities in positive characteristic. It can be generalized by focussing on the number of non-free direct summands. In this paper, we provide several methods to compute the (generalized) $F$-signature of a Hibi ring which is a special class of toric rings. In particular, we show that it can be computed by counting the elements in the symmetric group satisfying certain conditions. As an application, we also give the formula of the (generalized) $F$-signature for some Segre products of polynomial rings.