2020/06/30 by Myungho Kim, Doyun Koo
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Brauer group #Centralizer and normalizer #Combinatorics #Discrete mathematics #Invariant (physics) #Mathematical physics #Mathematics #Partition (number theory) #Permutation (music) #Permutation group #Pure mathematics #Quiver #Symmetric group #math.CO #math.RA #math.RT #msc:05A18 #msc:13A50 #msc:20G43
paper · pdf · doi:10.1016/j.jalgebra.2021.01.005
20 pages, changes of wrong conditions, typos, and grammar. Brauer algebras. Journal of Algebra (2021)
openalex created_date 2020/07/02 · openalex publication_date 2021/02/03 · arxiv created 2021/03/05 · arxiv updated 2021/03/08 · openalex updated_date 2026/08/05
We identify the dimension of the centralizer of the symmetric group \mathfrakSd in the partition algebra Ad(δ) and in the Brauer algebra Bd(δ) with the number of multidigraphs with d arrows and the number of disjoint union of directed cycles with d arrows, respectively. Using Schur-Weyl duality as a fundamental theory, we conclude that each centralizer is related with the G-invariant space Pd(Mn(k))G of degree d homogeneous polynomials on n × n matrices, where G is the orthogonal group and the group of permutation matrices, respectively. Our approach gives a uniform way to show that the dimensions of Pd(Mn(k))G are stable for sufficiently large n.