Polynomial Factorization Statistics and Point Configurations in ℝ3

Hyde, Trevor

doi:10.1093/imrn/rny271

Cited by 6 publications

(12 citation statements)

References 15 publications

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“…The proof of the liminal reciprocity for type polynomials (Theorem 1. The relationship between the liminal first moments of squarefree factorization statistics and representations of the symmetric group parallels our results in [9]. Church, Ellenberg, and Farb [6] established the connection between first moments of squarefree factorization statistics for univariate polynomials and the cohomology of point configurations in R 2 with their twisted Grothendieck-Lefschetz formula for squarefree polynomials.…”

Section: 2supporting

confidence: 81%

“…A function P defined on Poly d,n (F q ) is called a factorization statistic if P (f ) depends only on the factorization type of f . In [9] we found a surprising connection between the first moments of factorization statistics on the set of univariate polynomials (n = 1) and the cohomology of point configurations in Euclidean space viewed as a representation of the symmetric group. See Section 3 for precise definitions.…”

Section: 2mentioning

confidence: 95%

“…4.1]. The general polynomial case (1) was shown by the author [9] using different methods which also led to a new proof of the squarefree case. Theorem 1.3 provides a bridge between the arithmetic and combinatorics of factorization statistics on one hand and the geometry and representation theory of configuration spaces on the other.…”

Section: 2mentioning

confidence: 99%

“…The results in [9] are expressed in terms of expected values of factorization statistics. In this paper we focus on first moments as they lead to a cleaner statement for Theorem 1.4.…”

Section: 2mentioning

confidence: 99%

“…Equivalently, P is a function defined on the partitions of the degree d, or a class function of the symmetric group S d . In [9] we determined explicit formulas for the first moments of factorization statistics on Poly d,1 (F q ) and Poly sf d,1 (F q ) in terms of the characters of symmetric group representations related to the cohomology of point configurations in Euclidean space.…”

Section: Liminal First Moments Of Squarefree Factorization Statisticsmentioning

confidence: 99%

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Liminal reciprocity and factorization statistics

Hyde¹

2019

Algebraic Combinatorics

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Let M d,n (q) denote the number of monic irreducible polynomials in Fq[x1, x2, . . . , xn] of degree d. We show that for a fixed degree d, the sequence M d,n (q) converges coefficientwise to an explicitly determined rational function M d,∞ (q). The limit M d,∞ (q) is related to the classic necklace polynomial M d,1 (q) by an involutive functional equation we call liminal reciprocity. The limiting first moments of factorization statistics for squarefree polynomials are expressed in terms of symmetric group characters as a consequence of liminal reciprocity, giving a liminal analog of the twisted Grothendieck-Lefschetz formula of Church, Ellenberg, and Farb.

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Section: 2supporting

confidence: 81%

Section: 2mentioning

confidence: 95%

Section: 2mentioning

confidence: 99%

“…The results in [9] are expressed in terms of expected values of factorization statistics. In this paper we focus on first moments as they lead to a cleaner statement for Theorem 1.4.…”

Section: 2mentioning

confidence: 99%

Section: Liminal First Moments Of Squarefree Factorization Statisticsmentioning

confidence: 99%

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Liminal reciprocity and factorization statistics

Hyde¹

2019

Algebraic Combinatorics

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Configuration Spaces of Points: A User’s Guide

Kallel

2025

Encyclopedia of Mathematical Physics

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Eulerian representations for real reflection groups

Brauner

2022

Journal of London Math Soc

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The Eulerian idempotents, first introduced for the symmetric group and later extended to all reflection groups, generate a family of representations called the Eulerian representations that decompose the regular representation. In Type A$A$, the Eulerian representations have many elegant but mysterious connections to rings naturally associated with the braid arrangement. Here, we unify these results and show that they hold for any reflection group of coincidental type — that is, Sn$S_{n}$, Bn$B_{n}$, H3$H_{3}$ or the dihedral group I2(m)$I_{2}(m)$ — by giving six characterizations of the Eulerian representations, including as components of the associated graded Varchenko–Gelfand ring V$\mathcal {V}$. As a consequence, we show that Solomon's descent algebra contains a commutative subalgebra generated by sums of elements with a fixed number of descents if and only if W$W$ is coincidental. More generally, for any finite real reflection group, we give a case‐free construction of a family of Eulerian representations described by a flat decomposition of the ring V$\mathcal {V}$.

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Polynomial Factorization Statistics and Point Configurations in ℝ3

Cited by 6 publications

References 15 publications

Liminal reciprocity and factorization statistics

Liminal reciprocity and factorization statistics

Configuration Spaces of Points: A User’s Guide

Eulerian representations for real reflection groups

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