Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Webs and smooth components of two column Springer fibers

Mike Cummings

Abstract

Webs and Springer fibers are separately important objects in representation theory: webs give a diagrammatic calculus for tensor invariants of $\mathfrak{sl}_k$, and the cohomology group of Springer fibers can be used to construct the irreducible representations of the symmetric group. Fung's 1997 thesis gave the first evidence of a connection between $\mathfrak{sl}_2$ webs and Springer fibers, showing that webs naturally index and describe the components of certain "two row" Springer fibers. However, this case is known to be far from generic. This paper deepens this connection with a similar correspondence in the substantially more complicated "two column" case. In particular, and building on works of Fresse, Melnikov, and Sakas-Obeid, we use webs to give a clean characterization of the smooth components of two column rectangle Springer fibers and a simple description of the geometry of these smooth components. We also show that the Poincaré polynomial of the smooth components is invariant under the natural dihedral action on the corresponding webs.

Webs and smooth components of two column Springer fibers

Abstract

Webs and Springer fibers are separately important objects in representation theory: webs give a diagrammatic calculus for tensor invariants of , and the cohomology group of Springer fibers can be used to construct the irreducible representations of the symmetric group. Fung's 1997 thesis gave the first evidence of a connection between webs and Springer fibers, showing that webs naturally index and describe the components of certain "two row" Springer fibers. However, this case is known to be far from generic. This paper deepens this connection with a similar correspondence in the substantially more complicated "two column" case. In particular, and building on works of Fresse, Melnikov, and Sakas-Obeid, we use webs to give a clean characterization of the smooth components of two column rectangle Springer fibers and a simple description of the geometry of these smooth components. We also show that the Poincaré polynomial of the smooth components is invariant under the natural dihedral action on the corresponding webs.
Paper Structure (15 sections, 16 theorems, 24 equations, 11 figures)

This paper contains 15 sections, 16 theorems, 24 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Combinatorics of tableaux
  4. Springer fibers
  5. Geometry for two column rectangle Springer fibers
  6. Webs
  7. Degree two $\sl_k$ webs
  8. Geometry of smooth components
  9. Characterization of smooth components
  10. Geometry from degree two $\sl_k$ webs
  11. Poincaré polynomials for smooth components
  12. Geometry from noncrossing matching and ray diagrams
  13. Noncrossing matching and ray diagrams
  14. Characterization of smooth components
  15. Geometry from noncrossing matching and ray diagrams

Key Result

Theorem 1.1

The Springer fiber $S_\eta$ is equidimensional of dimension $\sum_{i\ge 1} (i-1)\eta_i$. Moreover, the components of $S_\eta$ are in bijective correspondence with the standard Young tableaux of shape $\eta$.

Figures (11)

  • Figure 1: Two column tableaux and their associated degree two $\sl_8$ webs.
  • Figure 2: The diagram of the two column rectangle $\eta = (2, 2, 2, 2) = (4, 4)^*$ and three standard Young tableaux of shape $\eta$.
  • Figure 3: Step-by-step construction of a noncrossing matching from the corresponding two column rectangular tableau.
  • Figure 4: Left-to-right: The weighted dissection, a weighted triangulation, and the interior of the resulting web (in green) corresponding to the noncrossing matching in \ref{['noncrossing construction']}.
  • Figure 5: The example from \ref{['noncrossing construction']} continued. Left: The web obtained from the triangulation in \ref{['weighted dissection']}. Right: The web obtained by choosing the other diagonal in the triangulation.
  • ...and 6 more figures

Theorems & Definitions (44)

  • Theorem 1.1: Spaltenstein (see also VargasvL)
  • Corollary : \ref{['dihedral invariance of smooth components']}
  • Example 1.3
  • Theorem 2.1: FMsingcomp
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4: FMSO
  • Theorem 2.5: FMSO
  • Theorem 2.7: FMSO
  • Definition 2.8
  • ...and 34 more