2025/07/30 by Grinberg, Darij · 1 citation
#05A05 #05E10 #20C30 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2507.22386
Let A be the group algebra k[Sn] of the n-th symmetric group Sn over a commutative ring k. For any two subsets A and B of [n], we define the elements ∇B,A:=∑_\substackw∈ Sn;
w( A) =B w and \widetilde∇B,A:=∑_\substackw∈ Sn;
w( A) ⊆ Bw of A. We study these elements, showing in particular that their minimal polynomials factor into linear factors (with integer coefficients). We express the product ∇D,C∇B,A as a ℤ-linear combination of ∇U,V's.
More generally, for any two set compositions (i.e., ordered set partitions) A and B of \ 1,2,…,n\ , we define ∇B,A\inA to be the sum of all permutations w∈ Sn that send each block of A to the corresponding block of B. This generalizes ∇B,A. The factorization property of minimal polynomials does not extend to the ∇B,A, but we describe the ideal spanned by the ∇B,A and a further ideal complementary to it. These two ideals have a "mutually annihilative" relationship, are free as k-modules, and appear as annihilators of tensor product Sn-representations; they are also closely related to Murphy's cellular bases, Specht modules, pattern-avoiding permutations and even some algebras appearing in quantum information theory.