A quotient of Fomin-Kirillov Algebra and q-Lucas polynomial

TL;DR

This paper introduces a new quotient of the Fomin-Kirillov algebra associated with cycle graphs, revealing its basis, dimension as Lucas numbers, and Hilbert series as q-Lucas polynomials, along with its symmetry properties.

Contribution

It defines a novel quotient algebra linked to cycle graphs, establishes its basis, dimension, Hilbert series, and character map, connecting algebraic structures with Lucas numbers and q-polynomials.

Findings

Basis corresponds to matchings in an n-cycle graph

Dimension equals Lucas number L_n

Hilbert series is the q-Lucas polynomial

Abstract

We introduce a quotient of Fomin-Kirillov algebra $FK(n)$ denoted $\overline{FK}_{C_n}(n)$, over the ideal generated by the edges of a complete graph on n vertexes that are missing in the $n$-cycle graph $C_n$. For this quotient algebra $\overline{FK}_{C_n}(n)$, we show that the basis is in one-to-one correspondence with the set of matchings in an $n$-cycle graph. We also prove that the dimension of $\overline{FK}_{C_n}(n)$ equals the Lucas Number $L_n$ and its Hilbert series is $q$-Lucas polynomial. We find the character map of this quotient algebra over Dihedral group $D_n$.