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Microlocal multiplicity of matroid Schubert varieties

Yiyu Wang

TL;DR

The paper introduces the microlocal multiplicity $m_M$ of matroid Schubert varieties, a new combinatorial invariant extracted from the characteristic cycle of the intersection complex. It provides an explicit formula $m_M = (-1)^d ext{rank}(M) ext{sum}_{F eq ext{empty}} 2^{ ext{rk}F} ext{χ}_{M^F}(1/2) ext{P}_{M_F}(1)$ and proves its invariance under matroid operations, together with a nonnegativity conjecture in the non-realizable case. By developing recursive relations for local Euler obstructions and closed expressions $ ext{Eu}_M = ext{χ}_M(2)$ and $c_M = -2^{ ext{rk}M} ext{χ}_M(1/2)$, the work links matroid combinatorics with geometric invariants via MacPherson and CSM/Chern-Mather theory. The microlocal multiplicity equation $(-1)^{ ext{rk}M}P_M(1)= abla m_{M_F}(-1)^{ ext{rk}F} ext{Eu}_{M^F}$ characterizes $m_M$ as the inverse in a specific Euler-obstruction matrix, and the multiplicativity and positivity results for realizable matroids underscore the interplay between combinatorics and singularity theory with potential extensions to arbitrary matroids.

Abstract

We study the multiplicity number of the characteristic cycle of the intersection complex of the matroid Schubert variety. It is shown to be a combinatorial invariant, and it can be computed by explicit formulas. We also conjecture that the generalization to arbitrary matroid is non-negative.

Microlocal multiplicity of matroid Schubert varieties

TL;DR

The paper introduces the microlocal multiplicity of matroid Schubert varieties, a new combinatorial invariant extracted from the characteristic cycle of the intersection complex. It provides an explicit formula and proves its invariance under matroid operations, together with a nonnegativity conjecture in the non-realizable case. By developing recursive relations for local Euler obstructions and closed expressions and , the work links matroid combinatorics with geometric invariants via MacPherson and CSM/Chern-Mather theory. The microlocal multiplicity equation characterizes as the inverse in a specific Euler-obstruction matrix, and the multiplicativity and positivity results for realizable matroids underscore the interplay between combinatorics and singularity theory with potential extensions to arbitrary matroids.

Abstract

We study the multiplicity number of the characteristic cycle of the intersection complex of the matroid Schubert variety. It is shown to be a combinatorial invariant, and it can be computed by explicit formulas. We also conjecture that the generalization to arbitrary matroid is non-negative.
Paper Structure (9 sections, 20 theorems, 71 equations)

This paper contains 9 sections, 20 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The geometry of matroid Schubert varieties
  4. Constructible sheaves, Constructible functions, Characteristic cycles and Local Euler obstructions
  5. Chern classes
  6. Chern-Schwartz-Macpherson classes of matroids
  7. Local Euler obstructions of matroid Schubert varieties
  8. Solving for the Euler obstruction number
  9. Mircolocal multiplicity

Key Result

Theorem 1.1

For a rank $d$ loopless matroid $M=M(\mathcal{A})$ realized by a hyperplane arrangement $\mathcal{A}$, the multiplicity of $\mathrm{CC}(\mathrm{IC}_{Y_\mathcal{A}})$ at the conormal variety to the all-infinity point is where $\mathcal{L}(M)$ denotes the set of all flats of $M$, $\chi_M$ is the characteristic polynomial of $M$, $P_M$ is the Kazhdan-Lusztig polynomial of $M$, $M^F$ and $M_F$ denote

Theorems & Definitions (38)

  • Theorem 1.1
  • Conjecture
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Proposition 2.1: ELIAS201636
  • Theorem 2.2: ELIAS201636
  • Proposition 2.3: MR3739192
  • Definition
  • Theorem 2.4: MR3999674
  • ...and 28 more