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Enumerating matrices with prescribed entries in an adjoint orbit

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Pith's one-line read A Hall scalar product of skew modified Hall-Littlewood and q-Whittaker functions counts extensions of a partial linear map to an endomorphism with prescribed similarity invariants over a finite field.

desk verdict The paper gives a Hall scalar product formula, in skew modified Hall-Littlewood and q-Whittaker functions, for counting extensions of a partial linear map with fixed columns to an endomorphism with prescribed similarity invariants over F_q. read the letter →

arxiv 2606.27497 v1 pith:VYFKN6FX submitted 2026-06-25 math.CO math.AGmath.RT

Samrith Ram This is my paper · ORCID
classification math.COmath.AGmath.RT
keywords adjointorbitsfinitefieldsHall-Littlewoodfunctionsq-WhittakerSmithnormalformmatrixpolynomialssimilarityinvariantsconjugacyclasses
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies matrices in a fixed adjoint orbit that also satisfy prescribed entries in complete columns. It supplies an explicit counting formula for the number of such matrices that additionally realize given similarity invariants. The formula is obtained as a scalar product and is applied to count monic matrix polynomials with fixed Smith normal form or fixed determinant, as well as to recover the classical count of matrices with a given characteristic polynomial.

What carries the argument

Hall scalar product formula for the number of extensions of a partially defined linear map to an endomorphism with prescribed similarity invariants, expressed via skew modified Hall-Littlewood and q-Whittaker functions.

What would settle it

Explicit enumeration, for a small prime power q and small matrix dimension, of all matrices extending a concrete partial map and realizing concrete invariants, compared against the numerical value of the proposed scalar product.

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Extended reading notes

Core claim

For a partially defined linear map over a finite field, the number of extensions to an endomorphism whose similarity invariants are prescribed is given by a Hall scalar product expressed in terms of skew modified Hall-Littlewood functions and q-Whittaker functions.

Load-bearing premise

The counting formula holds when the prescribed entries form complete columns and the similarity invariants are compatible with the Hall scalar product construction.

Editorial extensions

If this is right

  • The number of monic matrix polynomials over F_q with a prescribed Smith normal form is obtained directly from the formula.
  • The number of monic matrix polynomials over F_q with a prescribed determinant is obtained directly from the formula.
  • The Gerstenhaber-Reiner count of square matrices with a fixed characteristic polynomial is recovered as a special case.
  • Known point-count formulas for Hessenberg varieties yield related formulas for Hessenberg supports involving chromatic quasisymmetric functions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The scalar-product approach may extend to other partial-entry patterns once suitable symmetric-function identities are identified.
  • The appearance of chromatic quasisymmetric functions raises the question of whether the resulting generating functions remain polynomials for more general affine slices.
  • The same counting technique could be tested on related problems such as counting nilpotent matrices with prescribed entries and Jordan form.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The paper studies intersections of conjugacy classes of square matrices over finite fields with affine subspaces (prescribed entries). Its main result gives a Hall scalar product formula, in terms of skew modified Hall-Littlewood functions and q-Whittaker functions, for the number of extensions of a partial linear map (with prescribed complete columns) to an endomorphism having prescribed similarity invariants. Applications recover the Gerstenhaber-Reiner formula, count monic matrix polynomials with given Smith normal form or determinant, and relate Hessenberg supports to chromatic quasisymmetric functions.

Significance. If the central formula holds, the work supplies a new, explicit combinatorial tool for counting problems in linear algebra over finite fields that are governed by similarity invariants. The recovery of the Gerstenhaber-Reiner count and the applications to Smith normal forms of matrix polynomials constitute independent consistency checks. The link to Hessenberg varieties and chromatic quasisymmetric functions raises well-posed polynomiality questions for more general supports.

minor comments (2)
  1. The abstract is dense; a short sentence clarifying the precise compatibility conditions on the partial map and the invariants would help readers locate the main theorem.
  2. Notation for the Hall scalar product and the skew functions is introduced without an early reference to the relevant Macdonald or Hall-Littlewood literature; a single sentence directing the reader to the standard sources would improve accessibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the significance of the work, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation applies known Hall scalar product to counting problem

full rationale

The central result is a formula for the number of extensions of a partial linear map (prescribed complete columns) to an endomorphism with given similarity invariants, expressed via skew modified Hall-Littlewood and q-Whittaker functions. The manuscript recovers the independent Gerstenhaber-Reiner formula and gives applications to Smith normal forms of matrix polynomials. These external checks confirm the construction does not reduce to its inputs by definition or self-citation. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided description or abstract.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The paper applies existing symmetric-function machinery to a new enumeration setting and introduces no new free parameters or postulated entities.

assumptions (1)
  • standard math Standard algebraic and combinatorial properties of Hall-Littlewood and q-Whittaker functions, including the Hall scalar product
    The formula is expressed directly in terms of these functions and their scalar product.

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Pith. "Pith review of Enumerating matrices with prescribed entries in an adjoint orbit." pith.science (2026). https://pith.science/paper/VYFKN6FX

@misc{pith2026260627497,
  author       = {Pith},
  title        = {Pith review of: Enumerating matrices with prescribed entries in an adjoint orbit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYFKN6FX}},
  note         = {Machine review of arXiv:2606.27497}
}
abstract

We study intersections of conjugacy classes of square matrices over a finite field with affine coordinate subspaces, or equivalently matrices in a fixed adjoint orbit with prescribed entries. Our main result treats the case of prescribed columns: for a partially defined linear map we give a Hall scalar product formula for the number of extensions to an endomorphism with prescribed similarity invariants. This formula is expressed in terms of skew modified Hall--Littlewood functions and $q$-Whittaker functions. As applications, we count monic matrix polynomials over $\mathbb{F}_q$ with prescribed Smith normal form and with prescribed determinant, and recover the Gerstenhaber--Reiner formula for the number of square matrices with a fixed characteristic polynomial. We also note that known point-count formulas for Hessenberg varieties imply related formulas for Hessenberg supports involving chromatic quasisymmetric functions, motivating polynomiality questions for more general supports and prescribed affine slices.

Discussion (0). Continue with ORCID to comment.

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