Generalised spherical minors and their relations

TL;DR

This paper introduces generalized spherical minors for minimal rank spherical subgroups, extending classical minors and revealing new polynomial relations akin to exchange relations in LP-algebras.

Contribution

It defines generalized spherical minors for minimal rank spherical subgroups and proves they satisfy new polynomial relations extending classical identities.

Findings

Generalized spherical minors extend Fomin-Zelevinsky minors.

They satisfy polynomial relations with integer coefficients.

These relations resemble exchange relations in LP-algebras.

Abstract

Let H be a spherical subgroup of minimal rank of the semisimple simply connected complex algebraic group G. We define some functions on the homogeneous space G/H that we call generalised spherical minors. When G = H x H, we recover Fomin-Zelevinsky generalised minors. We prove that generalised spherical minors satisfy some integer coefficients polynomial relations, that extend the identities of classical generalised minors due to Fomin and Zelevinsky, and that have the shape of exchange relations of LP-algebras.