arXiv subjects

Rebecca L. Jayne

Publications and source records attributed to Rebecca L. Jayne.

4 recordsLinked to original sources

arXiv2022

Multiplicities of maximal weights of the $\hat{s\ell}(n) $-module $V(kΛ_0)$

Consider the affine Lie algebra $\hat{s\ell}(n)$ with null root $δ$, weight lattice $P$ and set of dominant weights $P^+$. Let $V(kΛ_0), \, k \in \mathbb{Z}_{\geq 1}$ denote the integrable highest weight $\hat{s\ell}(n)$-module with level $k \geq 1$ highest weight $kΛ_0$. Let $wt(V)$ denote the set of weights of $V(kΛ_0)$. A weight $μ\in wt(V)$ is a maximal weight if $μ+ δ\not\in wt(V)$. Let $max^+(kΛ_0)= max(kΛ_0)\cap P^+$ denote the set of maximal dominant weights which is known to be a finite set. In 2014, the authors gave the complete description of the set $max^+(kΛ_0)$. In subsequent papers the multiplicities of certain subsets of $max^+(kΛ_0)$ were given in terms of some pattern-avoiding permutations using the associated crystal base theory. In this paper the multiplicity of all the maximal dominant weights of the $\hat{s\ell}(n) $-module $V(kΛ_0)$ are given generalizing the known results.

math.RT↗

arXiv2020

Multiplicities of some maximal dominant weights of the $\widehat{s\ell}(n)$-modules $V(kΛ_0)$

For $n \geq 2$ consider the affine Lie algebra $\widehat{s\ell}(n)$ with simple roots $\{α_i \mid 0 \leq i \leq n-1\}$. Let $V(kΛ_0), \, k \in \mathbb{Z}_{\geq 1}$ denote the integrable highest weight $\widehat{s\ell}(n)$-module with highest weight $kΛ_0$. It is known that there are finitely many maximal dominant weights of $V(kΛ_0)$. Using the crystal base realization of $V(kΛ_0)$ and lattice path combinatorics we determine the multiplicities of a large set of maximal dominant weights of the form $kΛ_0 - λ^\ell_{a,b}$ where $ λ^\ell_{a,b} = \ellα_0 + (\ell-b)α_1 + (\ell-(b+1))α_2 + \cdots + α_{\ell-b} + α_{n-\ell+a} + 2α_{n - \ell+a+1} + \ldots + (\ell-a)α_{n-1}$, and $k \geq a+b$, $a,b \in \mathbb{Z}_{\geq 1}$, $\max\{a,b\} \leq \ell \leq \left \lfloor \frac{n+a+b}{2} \right \rfloor-1 $. We show that these weight multiplicities are given by the number of certain pattern avoiding permutations of $\{1, 2, 3, \ldots \ell\}$.

math.RT↗

arXiv2015

Lattice Paths, Young Tableaux, and Weight Multiplicities

For $\ell \geq 1$ and $k \geq 2$, we consider certain admissible sequences of $k-1$ lattice paths in a colored $\ell \times \ell$ square. We show that the number of such admissible sequences of lattice paths is given by the sum of squares of the number of standard Young tableaux of partitions of $\ell$ with height $\leq k$, which is also the number of $(k+1)k\cdots21$-avoiding permutations of $\{1, 2, \ldots, \ell\}$. Finally, we apply this result to the representation theory of the affine Lie algebra $\widehat{sl}(n)$ and show that this quantity gives the multiplicity of certain maximal dominant weights in the irreducible module $V(kΛ_0)$.

math.CO↗

arXiv2013

On multiplicities of maximal weights of $\hat{sl}(n)$-modules

We determine explicitly the maximal dominant weights for the integrable highest weight $\hat{sl}(n)$-modules $V((k-1)Λ_0 + Λ_s)$, $0 \leq s \leq n-1$, $ k \geq 2$. We give a conjecture for the number of maximal dominant weights of $V(kΛ_0)$ and prove it in some low rank cases. We give an explicit formula in terms of lattice paths for the multiplicities of a family of maximal dominant weights of $V(kΛ_0)$. We conjecture that these multiplicities are equal to the number of certain pattern avoiding permutations. We prove that the conjecture holds for $k=2$ and give computational evidence for the validity of this conjecture for $k >2$.

math.RT↗