Mirabolic Robinson-Schensted-Knuth correspondence

Roman Travkin

Mirabolic Robinson-Schensted-Knuth correspondence

Algebraic Geometry 2021-11-09 v4 Combinatorics

Authors: Roman Travkin

Abstract

The set of orbits of $GL(V)$ in $Fl(V)\times Fl(V)\times V$ is finite, and is parametrized by the set of certain decorated permutations in a work of Solomon. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in the bimodule over the Iwahori-Hecke algebra of $GL(V)$ arising from $Fl(V)\times Fl(V)\times V$ . We also give conjectural applications to the classification of unipotent mirabolic character sheaves on $GL(V)\times V$ .

Keywords

Mirabolic RSK correspondence decorated permutations standard Young tableaux combinatorial cells conormal vector Kazhdan-Lusztig cells Iwahori-Hecke algebra mirabolic character sheaves

Cite

@article{arxiv.0802.1651,
  title  = {Mirabolic Robinson-Schensted-Knuth correspondence},
  author = {Roman Travkin},
  journal= {arXiv preprint arXiv:0802.1651},
  year   = {2021}
}

Comments

28 pages. The list of conditions for the formulas for the action of generators on the mirabolic bimodule over the Hecke algebra corrected; the mirabolic RSK example modified; other minor changes. Document style reverted to amsart

Mirabolic Robinson-Schensted-Knuth correspondence

Abstract

Keywords

Cite

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