Temperley-Lieb Immanants of Ribbon Decomposition Matrices

TL;DR

This paper proves that certain algebraic elements called Temperley-Lieb immanants are Schur-positive on ribbon decomposition matrices, extending known results and conjecturing broader positivity in dual canonical bases.

Contribution

It establishes Schur-positivity of Temperley-Lieb immanants on ribbon matrices and conjectures this positivity for all dual canonical basis elements.

Findings

Temperley-Lieb immanants are Schur-positive on ribbon decomposition matrices.

This extends Haiman's result for Jacobi-Trudi matrices.

Conjecture that all dual canonical basis elements are Schur-positive on these matrices.

Abstract

Ribbon decomposition matrices give determinantal formulas for skew Schur functions that include as special cases the classical Jacobi-Trudi, Giambelli, and Lascoux-Pragacz formulas. We prove that certain elements of Lusztig's dual canonical basis, called Temperley-Lieb immanants, are Schur-positive when evaluated on ribbon decomposition matrices. We conjecture that this positivity holds for all elements of the dual canonical basis. This is known in the special case of Jacobi-Trudi matrices by a result of Haiman.