Fermionic Diagonal Coinvariants

Loading...
Thumbnail Image
Degree type
Doctor of Philosophy (PhD)
Graduate group
Mathematics
Discipline
Subject
Mathematics
Funder
Grant number
License
Copyright date
2022-10-05T20:22:00-07:00
Distributor
Related resources
Author
Kim, Jongwon
Contributor
Abstract

Let $W$ be a complex reflection group of rank $n$ acting on its reflection representation $V \cong \mb{C}^n$. The doubly graded action of $W$ on the exterior algebra $\wedge (V \oplus V^)$ induces an action on the quotient by the ideal generate by $W$-invariants with vanishing constant term $\FDR_W = \wedge (V \oplus V^) / \langle \wedge (V \oplus V^*)^W_{+} \rangle$. We describe the bi-graded $W$-module structure of $\FDR_W$ and introduce a variant of Motzkin paths that descends to the standard monomial basis of $\FDR_W$ with respect to certain term order. The top degree of $\FDR_W$ exhibits the Narayana refinement of Catalan numbers. When $W = S_n$, the symmetric group, $\FDR_{S_n} \cong R_{n,0,2}$, where $R_{n,0,2}$ is the special case of the Boson-Fermionic diagonal coinvariants with two sets of Fermionic variables. In this case, the $(i,j)$-th degree component is a difference of Kronecker product of two hook Schur functions. In addition we consider a module $M_{n,m}$ spanned by $m$-ary strings of length $n$. When $m = 2$, as a vector space, $M_{n,2} \cong \mb{C}[X_n] / \langle x_1^2, \ldots, x_n^2 \rangle$. The trivial component of $\dr_n \otimes M_{n,2}$ is a weighted sum of $q,t$-Narayana numbers which is a different $q,t$-Catalan number than the alternant of $\dr_n$. At $t = 1$, the trivial component equals the inversion generating function for $321$-avoiding permutations.

Advisor
Jim Haglund
Julia Hartmann
Date of degree
2022-01-01
Date Range for Data Collection (Start Date)
Date Range for Data Collection (End Date)
Digital Object Identifier
Series name and number
Volume number
Issue number
Publisher
Publisher DOI
Journal Issue
Comments
Recommended citation