Robinson-Schensted correspondence for party algebras

Abstract

In this paper, we construct a bijective proof of the identity \(n^k=\sum_{[\tilde{\lambda}] \in \Lambda_n^k} f^{[\tilde{\lambda}]} m_k^{[\tilde{\lambda}]}\), where \(m_k^{[\tilde{\lambda}]}\) is the multiplicity of the irreducible representation of \(\mathbb{Z}_r\left\langle S_n\right.\) module indexed by \([\tilde{\lambda}] \in \Lambda_n^k, f^{[\tilde{\lambda}]}\) is the degree of the corresponding representation indexed by \([\tilde{\lambda}] \in \Lambda_n^k\) and \(\Lambda_n^k=\left\{[\tilde{\lambda}] \vdash n\left|\sum_{i=1}^k i\right| \lambda^{(i)} \mid=k\right\}\). We give the proof of Robinson-Schensted correspondence for the party algebras which gives the bijective proof of party diagrams and the pairs of vacillating tableaux.