2020/04/29 by Rebecca L. Jayne, Kailash C. Misra, Jayne, Rebecca L. +1
Mathematics · #05A05 #05A17 #05E10 #17B10 #17B37 #17B67 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2004.14470
openalex publication_date 2020/04/29 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
For n ≥ 2 consider the affine Lie algebra \widehatsℓ(n) with simple roots \αi | 0 ≤ i ≤ n-1\. Let V(kΛ0), k ∈ ℤ≥ 1 denote the integrable highest weight \widehatsℓ(n)-module with highest weight kΛ0. It is known that there are finitely many maximal dominant weights of V(kΛ0). Using the crystal base realization of V(kΛ0) and lattice path combinatorics we determine the multiplicities of a large set of maximal dominant weights of the form kΛ0 - λ^ℓa,b where λ^ℓa,b = ℓα0 + (ℓ-b)α1 + (ℓ-(b+1))α2 + ⋯ + αℓ-b + αn-ℓ+a + 2αn - ℓ+a+1 + … + (ℓ-a)αn-1, and k ≥ a+b, a,b ∈ ℤ≥ 1, max\a,b\ ≤ ℓ ≤ \lfloor (n+a+b)/(2) \rfloor-1 . We show that these weight multiplicities are given by the number of certain pattern avoiding permutations of \1, 2, 3, … ℓ\.