Seminário de Análise Funcional, Estruturas Lineares e Aplicações

Involutions for symmetries of Littlewood-Richardson (LR) coefficients

The action of the dihedral group $\mathbb Z_2\times \mathfrak{ S}_3$ on LR coefficients is considered with the action of $\mathbb Z_2$ realized by the transposition of a partition. The action of $\mathbb Z_2\times\mathfrak{ S}_3$ carries a linear time action of a subgroup $H$ of index two, and a bijection which goes from $H$ into the other coset is difficult. LR coefficients are preserved in linear time by the action of $H$ whereas the other half symmetries consisting of commutativity and transposition symmetries are hidden. The latter are given by a remaining generator, the reversal involution or Schutzenberger involution, by which one is able to reduce in linear time the LR commuters and LR transposers to each other.

To pass from symmetries of LR (skew) tableaux to symmetries of companion Gelfand-Tsetlin (GT) patterns we build on the crystal action of the longest permutation in the symmetric group on an LR tableau, and Lascoux's double crystal graph structure on biwords. This analysis also affords an explicit bijection between two of the interlocking GT patterns in a hive.

This is based on a joint work with A. Conflitti and R. Mamede.

Organizadores actuais: Helena Mascarenhas, Ângela Mestre.