[Paper Review] Mirabolic Robinson-Schensted-Knuth correspondence

Roman Travkin|arXiv (Cornell University)|Feb 12, 2008

Advanced Algebra and Geometry4 citations

TL;DR

This paper introduces a mirabolic Robinson-Schensted-Knuth (RSK) correspondence that establishes a bijection between decorated permutations indexing GL(V)-orbits in Fl(V) × Fl(V) × V and triples consisting of two standard Young tableaux and a partition. The correspondence partitions these orbits into combinatorial cells, which are shown to coincide with microlocal two-sided cells defined by the type of general conormal vectors. The authors further conjecture that these cells align with bimodule Kazhdan-Lusztig cells in the Hecke algebra bimodule arising from the flag variety product with V, and propose applications to the classification of unipotent mirabolic character sheaves.

ABSTRACT

The set of orbits of $GL(V)$ in $Fl(V) imes Fl(V) imes V$ 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 $Fl(V) imes Fl(V) imes V$. We also give conjectural applications to the classification of unipotent mirabolic character sheaves on $GL(V) imes V$.

Motivation & Objective

• To classify GL(V)-orbits in Fl(V) × Fl(V) × V via a new combinatorial correspondence.
• To establish a bijection between these orbits and triples of standard Young tableaux and a partition, generalizing the classical RSK algorithm.
• To prove that this correspondence matches the partition of orbits by the type of general conormal vectors (microlocal cells).
• To conjecture that the same partition corresponds to bimodule Kazhdan-Lusztig cells in the Hecke algebra bimodule RN.
• To propose applications to the classification of unipotent mirabolic character sheaves on GL(V) × V.

Proposed method

• Define a mirabolic RSK correspondence as a bijection between decorated permutations in RBN and triples (T1, T2, θ), where T1, T2 are standard tableaux of shapes ν, ν′ with |ν| = |ν′| = N, and θ is a partition with ν ⊃ θ ⊂ ν′.
• Construct closed irreducible subvarieties of the conormal variety Z, indexed by triples (ν ⊃ θ ⊂ ν′), as images of closures of conormal bundles to orbits.
• Use the action of the Iwahori-Hecke algebra HN on the Grothendieck group of GL(V)-equivariant mixed Tate complexes on Fl(V) × Fl(V) × V to define a bimodule RN.
• Define bimodule Kazhdan-Lusztig cells in RN via the Kazhdan-Lusztig basis indexed by orbits in RBN.
• Introduce an involution F on RBN via the Fourier-Deligne transform, showing it preserves left, right, and bimodule cells.
• Conjecture that the mirabolic RSK output (ν, ν′, θ) classifies both microlocal and bimodule KL cells, and that the asymptotic bimodule over Jν is isomorphic to the matrix bimodule via the mirabolic RSK map.

Experimental results

Research questions

• RQ1 Does the mirabolic RSK correspondence classify GL(V)-orbits in Fl(V) × Fl(V) × V via triples of standard tableaux and a partition?
• RQ2 Do the microlocal two-sided cells—defined by the nilpotent type of general conormal vectors—coincide with the cells defined by the mirabolic RSK output (ν, ν′, θ)?
• RQ3 Is the partition of RBN into bimodule Kazhdan-Lusztig cells equivalent to the partition induced by the mirabolic RSK correspondence?
• RQ4 Can the asymptotic bimodule over Lusztig’s Jν for diagonal cells (ν = ν′) be realized as a matrix bimodule via the mirabolic RSK algorithm?
• RQ5 Do the irreducible constituents of the pushforward CH(F) of GL(V)-equivariant perverse sheaves on Fl(V) × Fl(V) × V correspond to unipotent mirabolic character sheaves indexed by (ν, θ)?

Key findings

• The mirabolic RSK correspondence provides a bijective correspondence between the set of decorated permutations RBN and the set of triples (T1, T2, θ) where T1, T2 are standard tableaux of shapes ν, ν′ with |ν| = |ν′| = N, and θ is a partition such that ν ⊃ θ ⊂ ν′.
• The partition of RBN into microlocal two-sided cells, defined by the nilpotent type of general conormal vectors to orbits, coincides exactly with the partition induced by the mirabolic RSK output (ν, ν′, θ).
• The involution F on RBN induced by the Fourier-Deligne transform preserves left, right, and bimodule Kazhdan-Lusztig cells.
• The bimodule KL cells in RN are conjectured to be partitioned precisely by the mirabolic RSK output (ν, ν′, θ), with ˜w and ˜w′ in the same cell iff they yield the same (ν, ν′, θ).
• The asymptotic bimodule Jν⊃θ⊂ν over Jν is conjectured to be isomorphic to the matrix bimodule MatSt(ν) via the mirabolic RSK map, with t˜w mapping to eT1,T2 where (T1, T2) are constructed from ˜w.
• The class of CH(˜H˜w) in the K-group of unipotent mirabolic Weil sheaves is conjectured to be a linear combination of Fλ,μ with coefficients fλ,μ that vanish outside the summands corresponding to (ν, θ) = Υ(λ, μ).