Canonical bases and new applications of increasing and decreasing subsequences to invariant theory

TL;DR

This paper explores the use of canonical bases derived from Kazhdan--Lusztig and partition algebras to analyze invariants in tensor space, revealing integral dependencies and combinatorial rules for minimal entry sets.

Contribution

It introduces new applications of canonical bases to invariant theory, providing combinatorial methods to identify minimal invariant components.

Findings

Linear decomposition of invariants depends integrally on entries.

Combinatorial rules identify minimal sets of entries for invariants.

Canonical bases facilitate understanding of invariant structures.

Abstract

In 2012 Raghavan, Samuel, and Subrahmanyam showed that the Kazhdan--Lusztig basis for the Iwahori--Hecke algebra in type A provides a ``canonical'' basis for the centraliser algebra of the Schur algebra acting on tensor space. In 2022 the second author found a similar result for the centraliser of the partition algebra acting on the same tensor space. Each basis is indexed by permutations. We exploit these bases to show that the linear decomposition of an arbitrary invariant (in either centraliser algebra) depends integrally on its entries, and describe combinatorial rules that pick out minimal sets of such entries.