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

Abstract

The partition algebra $\mathsf {P}_k(n)$ and the symmetric group $\mathsf {S}_n$ are in Schur–Weyl duality on the k-fold tensor power $\mathsf {M}_n^{\otimes k}$ of the permutation module $\mathsf {M}_n$ of $\mathsf {S}_n$, so there is a surjection $\mathsf {P}_k(n) \rightarrow \mathsf {Z}_k(n) := \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})$, which is an isomorphism when $n \ge 2k$. We prove a dimension formula for the irreducible modules of the centralizer algebra $\mathsf {Z}_k(n)$ in terms of Stirling numbers of the second kind. Via Schur–Weyl duality, these dimensions equal the multiplicities of the irreducible $\mathsf {S}_n$-modules in $\mathsf {M}_n^{\otimes k}$. Our dimension expressions hold for any $n \ge 1$ and $k\ge 0$. 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 $\mathsf {M}_n^{\otimes k}$ and the quasi-partition algebra corresponding to tensor powers of the reflection representation of $\mathsf {S}_n$.

Appendices: Bratteli Diagrams

1.1 Appendix 1: Levels $\ell = 0,\frac{1}{2},1,\ldots ,\frac{7}{2},4$   of the Bratteli diagram $\mathcal {B}(\mathsf {S}_6,\mathsf {S}_5)$

Level $\ell = \frac{7}{2}$ is the first time the centralizer algebra loses a dimension from the generic dimension, which is the 7th Bell number $\mathsf {B}(7) = 877$.

1.2 Appendix 2: Levels $\ell = 0,\frac{1}{2},1,\ldots ,\frac{7}{2},4$ of the quasi-Bratteli diagram $\mathcal {QB}(\mathsf {S}_6,\mathsf {S}_5)$

To calculate the subscripts on the half-integer rows, use Pascal addition of the subscripts from the row above. To calculate the subscripts on integer level rows, first use Pascal addition from the row above, and then subtract the subscript on the same partition from two rows above.

1.3 Appendix 3: Levels $\ell = 0,\frac{1}{2},1,\ldots ,\frac{7}{2},4$ of the Bratteli diagram $\mathcal {B}(\mathsf {A}_6, \mathsf {A}_5)$

1.4 Appendix 4: Levels $\ell = 0,\frac{1}{2},1,\ldots ,\frac{7}{2},4$ of the quasi-Bratteli diagram $\mathcal {QB}(\mathsf {A}_6,\mathsf {A}_5)$

To calculate the subscripts on the half-integer rows, use Pascal addition of the subscripts from the row above. To calculate the subscripts on integer level rows, first use Pascal addition from the row above, and then subtract the subscript on the same partition from two rows above.