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Combinatorial proofs of the type A quiver component formulas

Aidan Lindberg, Jenna Rajchgot

Abstract

The K-theoretic quiver component formula expresses the K-polynomial of a type A quiver locus as an alternating sum of products of double Grothendieck polynomials. This formula was conjectured by A. Buch and R. Rimányi and later proved by R. Kinser, A. Knutson, and the second author. We provide a new proof of this formula which replaces Gröbner degenerations by combinatorics. Along the way, we obtain a new proof of A. Buch and R. Rimányi's cohomological quiver component formula. Again, our proof replaces geometric techniques by combinatorics.

Combinatorial proofs of the type A quiver component formulas

Abstract

The K-theoretic quiver component formula expresses the K-polynomial of a type A quiver locus as an alternating sum of products of double Grothendieck polynomials. This formula was conjectured by A. Buch and R. Rimányi and later proved by R. Kinser, A. Knutson, and the second author. We provide a new proof of this formula which replaces Gröbner degenerations by combinatorics. Along the way, we obtain a new proof of A. Buch and R. Rimányi's cohomological quiver component formula. Again, our proof replaces geometric techniques by combinatorics.

Paper Structure

This paper contains 15 sections, 23 theorems, 57 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Permutations and partial permutations
  4. Lacing diagrams
  5. Block structures on $d\times d$ matrices and the bipartite Zelevinsky permutation
  6. Pipe dreams
  7. Permutations and laces
  8. Relating pipes and laces
  9. Schubert and Grothendieck polynomials in terms of pipe dreams
  10. The bipartite codimension formula and the bipartite pipe formulas
  11. A combinatorial proof of the bipartite cohomological component formula
  12. A combinatorial proof of the bipartite $K$-theoretic component formula
  13. A bijection on sets of pipe dreams
  14. Bijection from $X_{\Omega}$ to $K$-theoretic lacing diagrams
  15. Proof of the bipartite type $A$ $K$-theoretic quiver component formula

Key Result

Lemma 2.6

Let $w$ be a $k \times \ell$ partial permutation matrix and let $c(w) \in S_m$ be its minimal length completion. Then, the extension via elbow tiles map gives us identifications Moreover, if $v \in S_{m'}$ is any extension of $c(w)$ by the identity permutation as in Remark , then $\mathop{\mathrm{RPipes}}\nolimits(w)=\mathop{\mathrm{RPipes}}\nolimits(v)$ and $\mathop{\mathrm{Pipes}}\nolimits(w)=

Figures (5)

  • Figure 1: A lacing diagram for quiver (1.1) and its associated extended lacing diagram.
  • Figure 2: Two $K$-theoretic lacing diagrams obtained from the minimal lacing diagram from Figure .
  • Figure 3: The Zelevinsky permutation matrix associated to the lacing diagram $\mathbf{w}$ from Figure .
  • Figure 4: $P_1,P_2\in \mathop{\mathrm{Pipes}}\nolimits(v_0,v(\Omega))$, where $v(\Omega)$ is the permutation from Figure . $P_1$ is reduced while $P_2$ is not. $P_*$ is defined above. The snake region is outlined with bold black lines.
  • Figure 5: The Zelevinsky Permutation associated to the lacing diagram $\mathbf{w}$ in Figure with the block structure now labeled by our alphabets $\mathbf{s}, \mathbf{t}$.

Theorems & Definitions (53)

  • Remark 2.1
  • Example 2.2
  • Example 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.9
  • Remark 2.10
  • Example 2.11
  • Proposition 2.12
  • proof
  • ...and 43 more