Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

• Published in: Journal of algebraic combinatorics 2017-08, Vol.46 (1), p.77-108
• Main Authors: Benkart, Georgia, Halverson, Tom, Harman, Nate
• Format: Article
• Language: English
• Subjects: Algebra; Analogs; Combinatorial analysis; Combinatorics; Computer Science; Convex and Discrete Geometry; Group theory; Group Theory and Generalizations; Isomorphism; Lattices; Mathematics; Mathematics and Statistics; Modules; Order; Ordered Algebraic Structures; Reciprocity; Reflection; Subgroups
• ISSN: 0925-9899; 1572-9192
• DOI: 10.1007/s10801-017-0748-4

The partition algebra P k ( n ) and the symmetric group S n are in Schur–Weyl duality on the k -fold tensor power M n ⊗ k of the permutation module M n of S n , so there is a surjection P k ( n ) → Z k ( n ) : = End S n ( M n ⊗ k ) , which is an isomorphism when n ≥ 2 k . We prove a dimension formula for the irreducible modules of the centralizer algebra Z k ( n ) in terms of Stirling numbers of the second kind. Via Schur–Weyl duality, these dimensions equal the multiplicities of the irreducible S n -modules in M n ⊗ k . Our dimension expressions hold for any n ≥ 1 and k ≥ 0 . Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on M n ⊗ k and the quasi-partition algebra corresponding to tensor powers of the reflection representation of S n .