Mirabolic Robinson-Schensted-Knuth correspondence
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 $GL(V)\times V$.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2008
- DOI:
- 10.48550/arXiv.0802.1651
- arXiv:
- arXiv:0802.1651
- Bibcode:
- 2008arXiv0802.1651T
- Keywords:
-
- Mathematics - Algebraic Geometry;
- Mathematics - Combinatorics
- E-Print:
- 28 pages. The list of conditions for the formulas for the action of generators on the mirabolic bimodule over the Hecke algebra corrected