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Operadic Kazhdan-Lusztig-Stanley theory

Basile Coron

TL;DR

This work introduces $oldsymbol{ ext{P}}$-operads as a unifying operadic framework indexed by finite bounded posets, linking poset topology with operadic algebra to categorize Kazhdan–Lusztig–Stanley (KLS) polynomials for geometric lattices. By developing Gröbner bases, bar constructions, and Koszul duality for $oldsymbol{ ext{P}}$-operads, the authors prove Koszulness of the Gerstenhaber-type operad on geometric lattices and construct KLS complexes inside the bar construction that recover KLS polynomials as Euler characteristics. The main achievement is a categorification of right/left/inverse KLS polynomials via the complexes $ extbf{RKLS}, extbf{LKLS}, extbf{RKLS}, extbf{LKLS}$, with precise cohomology concentration results and a new proof of positivity for geometric-lattice Kazhdan–Lusztig polynomials. Overall, the operadic perspective provides a robust, algebraic route to KLS theory that suggests further equivariant, Hodge-theoretic, and geometric connections beyond realizable cases.

Abstract

We introduce a new type of operad-like structure called a P-operad, which depends on the choice of some collection of posets P, and which is governed by chains in posets of P. We introduce several examples of such structures which are related to classical poset theoretic notions such as poset homology, Cohen--Macaulayness and lexicographic shellability. We then show that P-operads form a satisfactory framework to categorify Kazhdan--Lusztig polynomials of geometric lattices and their kernel. In particular, this leads to a new proof of the positivity of the coefficients of Kazhdan--Lusztig polynomials of geometric lattices.

Operadic Kazhdan-Lusztig-Stanley theory

TL;DR

This work introduces -operads as a unifying operadic framework indexed by finite bounded posets, linking poset topology with operadic algebra to categorize Kazhdan–Lusztig–Stanley (KLS) polynomials for geometric lattices. By developing Gröbner bases, bar constructions, and Koszul duality for -operads, the authors prove Koszulness of the Gerstenhaber-type operad on geometric lattices and construct KLS complexes inside the bar construction that recover KLS polynomials as Euler characteristics. The main achievement is a categorification of right/left/inverse KLS polynomials via the complexes , with precise cohomology concentration results and a new proof of positivity for geometric-lattice Kazhdan–Lusztig polynomials. Overall, the operadic perspective provides a robust, algebraic route to KLS theory that suggests further equivariant, Hodge-theoretic, and geometric connections beyond realizable cases.

Abstract

We introduce a new type of operad-like structure called a P-operad, which depends on the choice of some collection of posets P, and which is governed by chains in posets of P. We introduce several examples of such structures which are related to classical poset theoretic notions such as poset homology, Cohen--Macaulayness and lexicographic shellability. We then show that P-operads form a satisfactory framework to categorify Kazhdan--Lusztig polynomials of geometric lattices and their kernel. In particular, this leads to a new proof of the positivity of the coefficients of Kazhdan--Lusztig polynomials of geometric lattices.
Paper Structure (13 sections, 19 theorems, 90 equations, 3 figures)

This paper contains 13 sections, 19 theorems, 90 equations, 3 figures.

Table of Contents

  1. Introduction
  2. $\mathcal{P}$-operads: first definitions and examples
  3. Gröbner bases for $\mathcal{P}$-operads
  4. First definitions
  5. Main results about Gröbner bases
  6. Koszulness of $\mathcal{P}$-operads
  7. Bar construction
  8. Koszul dual
  9. Koszul complexes
  10. Gröbner bases and Koszul duality
  11. Application to Kazhdan--Lusztig--Stanley theory
  12. Reminders
  13. Categorification of KLS theory

Key Result

Theorem 1.2

Let $P$ be a locally finite graded bounded poset $P$ and $\kappa$ a $P$-kernel. There exists a unique collection of polynomials $(f_{XY})_{X\leq Y \in P}$ (resp. $(g_{XY})_{X\leq Y \in P}$) such that we have

Figures (3)

  • Figure 1: A generic lattice path associated to a summand in $\mathrm{gr}\,^{M, \alpha}\mathrm{\mathbf{B}}(\mathbf{Gerst})$.
  • Figure 2: The first page of the spectral sequence associated to $\mathcal{F}'_{|\mathbf{RKLS}}$ in the case $k-2i = 1$.
  • Figure 3: The first page of the spectral sequence associated to $\mathcal{F}'$ in half weight.

Theorems & Definitions (68)

  • Definition 1.1: P-kernel, Stanley_1992
  • Theorem 1.2: Stanley_1992, Corollary 6.7
  • Theorem 1.3: Theorem \ref{['theomainhomo']}
  • Definition 2.1: $\mathcal{P}$-collection, $\mathcal{P}$-operad
  • Example 2.2
  • Definition 2.3: Morphism between $\mathcal{P}$-collections/operads
  • Definition 2.4: Free operad generated by a collection
  • Proposition 2.5
  • Definition 2.6: Operadic ideal, ideal generated by a subcollection
  • Example 2.7
  • ...and 58 more