Enumerating traceless matrices over compact discrete valuation rings

Carnevale, Angela; Shechter, Shai; Voll, Christopher

doi:10.1007/s11856-018-1755-4

Cited by 11 publications

(25 citation statements)

References 16 publications

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“…Such results were first recorded by Stasinski and Voll [31]. For further related results, see [5,6] and references therein.…”

Section: Motivation: Shapes Ranks and Average Sizes Of Kernels Of Mat...supporting

confidence: 60%

Linear relations with disjoint supports and average sizes of kernels

Carnevale¹,

Rossmann²

2020

Preprint

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We study the effects of imposing linear relations within modules of matrices on average sizes of kernels. The relations that we consider can be described combinatorially in terms of partial colourings of grids. The cells of these grids correspond to positions in matrices and each defining relation involves all cells of a given colour. We prove that imposing such relations arising from "admissible" partial colourings has no effect on average sizes of kernels over finite quotients of discrete valuation rings. This vastly generalises the known fact that average sizes of kernels of general square and traceless matrices of the same size coincide over such rings. As a group-theoretic application, we explicitly determine zeta functions enumerating conjugacy classes of finite p-groups derived from free class-3-nilpotent groups for p 5.

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“…Such results were first recorded by Stasinski and Voll [31]. For further related results, see [5,6] and references therein.…”

Section: Motivation: Shapes Ranks and Average Sizes Of Kernels Of Mat...supporting

confidence: 60%

Linear relations with disjoint supports and average sizes of kernels

Carnevale¹,

Rossmann²

2020

Preprint

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“…We note that permutation statistics have previously featured in explicit formulae for representation zeta functions ; see also .…”

Section: Full Matrix Algebras Classical Lie Algebras and Relativesmentioning

confidence: 81%

“…Let

{prefixM}_{d}^{rk = i} false(F_{q} false) : = false{a \in M_{d} ({boldF}_{q}) : prefixrk (a) = i false}

. As explained in [, § 3.2], for

i = 0, \dots, d

0true (||, M_{d}^{prefixrk = d - i} ({boldF}_{q})) = q^{d^{2} - i^{2}} \sum_{σ \in B_{d}^{{i}^{c}}} {false(- 1 false)}^{prefixN false(σ false)} q^{- prefixlen false(σ false)}

whence

\begin{matrix} true \\ right prefixask false({prefixM}_{d} false(F_{q} false) false) & left = q^{- d^{2}} \sum_{i = 0}^{d} |{prefixM}_{d}^{rk = d - i} false(F_{q} false)| q^{i} \\ left = \sum_{i = 0}^{d} q^{i - i^{2}} \sum_{σ \in {prefixB}_{d}^{{false{i false}}^{c}}} {(- 1)}^{N (σ)} q^{- len (σ)} . \end{matrix}

…”

Section: Elementary Properties Of Average Sizes Of Kernelsunclassified

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The average size of the kernel of a matrix and orbits of linear groups

Rossmann

2018

Proc. London Math. Soc.

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Let frakturO be a compact discrete valuation ring of characteristic 0. Given a module M of matrices over frakturO, we study the generating function encoding the average sizes of the kernels of the elements of M over finite quotients of frakturO. We prove rationality and establish fundamental properties of these generating functions and determine them explicitly for various natural families of modules M. Using p‐adic Lie theory, we then show that special cases of these generating functions enumerate orbits and conjugacy classes of suitable linear pro‐p groups.

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“…Recently, permutation statistics have found applications to various zeta functions in algebra; see, for instance, [2,5,7,20,21]. An early instance of such applications arose from the enumeration of ideals in hereditary orders encoded in so-called genus zeta functions.…”

Section: Introductionmentioning

confidence: 99%

On Denert's statistic

Carnevale¹,

Tielker²

2021

Preprint

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We show that the numerators of genus zeta function associated with local hereditary orders studied by Denert can be described in terms of the joint distribution of Euler-Mahonian statistics on multiset permutations defined by Han. We use this result to deduce a reciprocity property for genus zeta functions of local hereditary orders whose associated composition is a rectangle. We also record a remarkable identity satisfied by genus zeta functions of local hereditary orders in terms of Hadamard products of genus zeta functions of maximal orders. Finally, we define Mahonian companions of excedance statistics on groups of signed and even-signed permutations.

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Enumerating traceless matrices over compact discrete valuation rings

Cited by 11 publications

References 16 publications

Linear relations with disjoint supports and average sizes of kernels

Linear relations with disjoint supports and average sizes of kernels

The average size of the kernel of a matrix and orbits of linear groups

On Denert's statistic

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