[Paper Review] Excursions into Algebra and Combinatorics at $q=0$

Tom Denton|arXiv (Cornell University)|Aug 22, 2011

Advanced Combinatorial Mathematics67 references3 citations

TL;DR

This paper investigates the algebraic and combinatorial structures of the 0-Hecke algebra and $χ$-trivial monoids at $q=0$, deriving a formula for orthogonal idempotents and establishing connections to crystal bases and pattern avoidance. It provides a computational implementation of Stembridge’s local axioms for simply-laced crystal bases, offering a structural characterization of finite-dimensional crystals for $U'_q(\hat{sl}_2)$.

ABSTRACT

We explore combinatorics associated with the degenerate Hecke algebra at $q=0$, obtaining a formula for a system of orthogonal idempotents, and also exploring various pattern avoidance results. Generalizing constructions for the 0-Hecke algebra, we explore the representation theory of $\JJ$-trivial monoids. We then discuss two-tensors of crystal bases for $U_q( ilde{\mathfrak{sl}_2})$, establishing a complementary result to one of Bandlow, Schilling, and Thiéry on affine crystals arising from promotion operators. Finally, we give a computer implementation of Stembridge's local axioms for simply-laced crystal bases.

Motivation & Objective

• To develop a systematic understanding of the representation theory of the 0-Hecke algebra and $χ$-trivial monoids at $q=0$.
• To generalize constructions of orthogonal idempotents in the 0-Hecke algebra to broader classes of monoids.
• To establish connections between non-decreasing parking functions, pattern avoidance, and combinatorial quotients in symmetric and affine types.
• To provide a computational framework for verifying the Stembridge local axioms for crystal bases in simply-laced types.
• To classify finite-dimensional crystals for $U'_q(\hat{sl}_2)$ using promotion operators and extremal vectors.

Proposed method

• Derives a formula for a system of orthogonal idempotents in the 0-Hecke algebra $H_0(S_N)$ using diagram demipotents and branching rules.
• Applies representation theory of $χ$-trivial monoids to analyze simple modules, radical, and Cartan matrices, showing they are upper uni-triangular.
• Introduces combinatorial quotients related to non-decreasing parking functions (NDPF) and establishes connections to pattern avoidance in symmetric and affine types.
• Uses Stembridge’s local axioms to characterize crystal bases, implementing checks via the `stembridgeTriple` and `_test_stembridge_local_axioms` functions in Sage.
• Employs tensor product structures and evaluation modules to analyze extremal vectors and promotion operators in $U'_q(\hat{sl}_2)$-crystals.
• Develops a computer implementation in Sage that verifies the Stembridge local axioms on finite crystal graphs, supporting both verbose and complete failure reporting.

Experimental results

Research questions

• RQ1 What is the structure of orthogonal idempotents in the 0-Hecke algebra of the symmetric group?
• RQ2 How can the representation theory of $χ$-trivial monoids be generalized beyond the 0-Hecke algebra?
• RQ3 What is the relationship between non-decreasing parking functions and pattern-avoiding permutations in types A, B, and affine A?
• RQ4 How do promotion operators and extremal vectors classify finite-dimensional crystals for $U'_q(\hat{sl}_2)$?
• RQ5 Can Stembridge’s local axioms be algorithmically verified for crystal bases in simply-laced types?

Key findings

• A closed-form formula is derived for a system of orthogonal idempotents in the 0-Hecke algebra $H_0(S_N)$, generalizing known constructions.
• The Cartan matrix of the monoid of order-preserving regressive functions on a poset $P$ is shown to be upper uni-triangular, confirming structural properties of $χ$-trivial monoids.
• A combinatorial quotient of the 0-Hecke algebra is identified with the set of non-decreasing parking functions, linking algebraic idempotents to pattern-avoiding permutations.
• The implementation of Stembridge’s local axioms in Sage correctly verifies the axioms for finite crystals of type $A_2$ and $B_2$, with examples confirming the triple condition $A_{ij} = b + c$.
• For crystals of type $A_3$, the `_test_stembridge_local_axioms` function successfully validates the axioms across all elements, with verbose output identifying no failures.
• The code correctly detects violations of axioms P3–P6, including cases where $\text{depth}(i,j) = -1$ and $\text{rise}(i,j) = -1$, ensuring structural consistency.