FFLV-type monomial bases for type $B$

TL;DR

This paper constructs combinatorial monomial bases for irreducible modules of the Lie algebra fso_{2n+1}, extending FFLV bases from types A and C, using Dyck paths with root-dependent weights.

Contribution

It introduces a new family of monomial bases for fso_{2n+1} modules, defined via weighted Dyck paths, expanding the combinatorial framework of FFLV bases.

Findings

Bases induce in certain degenerations of modules

Weighted Dyck path sums define the bases

Extension of FFLV bases to type B

Abstract

We present a combinatorial monomial basis (or, more precisely, a family of monomial bases) in every finite-dimensional irreducible $\mathfrak{so}_{2n+1}$-module. These bases are in many ways similar to the FFLV bases for types $A$ and $C$. They are also defined combinatorially via sums over Dyck paths in certain triangular grids. Our sums, however, involve weights depending on the length of the corresponding root. Accordingly, our bases also induce bases in certain degenerations of the modules but these degenerations are obtained not from the filtration by PBW degree but by a weighted version thereof.