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Equivariant Euler characteristics on permutohedral varieties

Vincenzo Galgano, Hanieh Keneshlou, Mateusz Michalek

Abstract

By the work of J.Huh, one can interpret binomial coefficients as a solution to an intersection problem on a permutohedral variety $X_E$. Applying Hirzebruch-Riemann-Roch, this intersection problem is equivalent to computing Euler characteristic of a specific element of $K$-theory of $X_E$. This element has a natural lifting to equivariant $K$-theory and thus the Euler characteristic may be upgraded to a Laurent polynomial. We provide and implement three different approaches, in particular a recursive one, to computing these polynomials.

Equivariant Euler characteristics on permutohedral varieties

Abstract

By the work of J.Huh, one can interpret binomial coefficients as a solution to an intersection problem on a permutohedral variety . Applying Hirzebruch-Riemann-Roch, this intersection problem is equivalent to computing Euler characteristic of a specific element of -theory of . This element has a natural lifting to equivariant -theory and thus the Euler characteristic may be upgraded to a Laurent polynomial. We provide and implement three different approaches, in particular a recursive one, to computing these polynomials.
Paper Structure (11 sections, 12 theorems, 91 equations, 1 figure)

This paper contains 11 sections, 12 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Permutohedral varieties and special divisors
  4. Permutohedral varieties.
  5. Divisors on the permutohedral variety.
  6. Equivariant localization formula
  7. Example of projective line
  8. Computation of the Euler characteristic
  9. First approach: computing cohomology groups
  10. Second approach: equivariant localization
  11. Recursive formula for Euler characteristic

Key Result

Lemma 2.1

For each $k,l\geq 0$, not both equal to zero, the divisor $k\Gamma_i + \ell \Delta_j\in \mathop{\mathrm{Div}}\nolimits^T(X_n)$ is nef and big. In particular, the anticanonical divisor $-K$ is big and nef.

Figures (1)

  • Figure :

Theorems & Definitions (26)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Theorem 2.3: tautologicalclasses, Theorem 2.1
  • Theorem 2.4: localization, 4.7
  • Theorem 3.1: Toric Kawamata-Viehweg; Theorem 9.3.10, cox
  • Theorem 3.2: Batyrev-Borisov vanishing; Theorem 9.2.7, cox
  • Proposition 3.3
  • proof
  • Remark 3.4
  • ...and 16 more