Character factorizations for representations of GL(n,C)

Chayan Karmakar

arxiv: 2210.03544 · v3 · submitted 2022-10-07 · 🧮 math.RT · math.GR

Character factorizations for representations of GL(n,C)

Chayan Karmakar This is my paper

Pith reviewed 2026-05-24 10:50 UTC · model grok-4.3

classification 🧮 math.RT math.GR

keywords character factorizationirreducible representationsgeneral linear groupWeyl groupcombinatorial cancellationLittlewood-RichardsonPrasad theorem

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The pith

The character of an irreducible representation of GL(mn,C) at diagonal elements with eigenvalues ω_n^{j-1} t_i equals the product of certain characters for GL(m,C) evaluated at diag(t_1^n, ..., t_m^n).

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

The paper supplies a direct combinatorial proof of a factorization theorem for characters of irreducible representations of GL(mn,C). It shows that the character evaluated at diagonal matrices whose eigenvalues cycle through the t_i multiplied by successive powers of a primitive nth root of unity factors as a product of the corresponding characters of GL(m,C) taken at the vector whose entries are the t_i each raised to the nth power. A sympathetic reader would care because the relation supplies an explicit bridge between representations of a larger matrix group and those of a smaller one, evaluated at specially chosen points. The argument proceeds by cancellation inside the Weyl group rather than by manipulating determinantal expressions.

Core claim

For integers m,n ≥ 2, the character of an irreducible representation of GL(mn,C) at diagonal elements with eigenvalues ω_n^{j-1} t_i (1≤i≤m, 1≤j≤n) equals the product of certain characters for GL(m,C) evaluated at diag(t_1^n, ..., t_m^n). The equality is established by a direct combinatorial cancellation argument within the Weyl group that removes all non-contributing terms.

What carries the argument

The direct combinatorial cancellation argument inside the Weyl group that eliminates non-contributing terms from the character sum.

If this is right

The factorization holds for every irreducible representation of GL(mn,C).
The same numerical identity is recovered whether one uses the Weyl-group cancellation or earlier determinantal methods.
The result specializes to the classical Littlewood-Richardson and Prasad theorems when the parameters m and n are fixed.
The identity is valid at every choice of the parameters t_i inside the torus.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same cancellation pattern may produce analogous factorizations when the evaluation points are replaced by other torsion elements of the torus.
The technique offers a template that could be tried on character formulas for other classical groups such as orthogonal or symplectic groups.
Direct computer checks for small m and n would give an independent test that every extraneous term has indeed cancelled.

Load-bearing premise

The combinatorial cancellation argument within the Weyl group accounts for all non-contributing terms without omissions or over-cancellations.

What would settle it

An explicit character computation for m=2, n=2 and any fixed irreducible representation of GL(4,C), evaluated at the indicated diagonal point, that fails to equal the predicted product of two GL(2,C) characters at diag(t_1^2, t_2^2).

read the original abstract

We give another proof of a theorem of D. Prasad (Theorem 2, \textit{Israel J. Math.} 2016), which is also a classical result of Littlewood--Richardson (Theorem VI, \textit{Q. J. Math.} 1934). For integers $m,n \ge 2$, this result calculates the character of an irreducible representation of $\GL(mn,\C)$ at diagonal elements with eigenvalues $\omega^{j-1}_nt_i$ for $1 \le i \le m$, $1 \le j \le n$, where $\omega_n=e^{2\pi \imath/n}$, expressing it as a product of certain characters for $\GL(m,\C)$ evaluated at $\underline{t}^n={\rm diag}(t_1^{n},t_{2}^{n},\dots,t_{m}^{n})$. Unlike previous approaches that rely on determinantal identities, our proof utilizes a direct combinatorial cancellation argument within the Weyl group.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a re-proof of a known Littlewood-Richardson/Prasad factorization using Weyl-group cancellation instead of determinants.

read the letter

The paper re-proves an existing result on the character of an irreducible representation of GL(mn,C) evaluated at points with eigenvalues that are n-th roots of unity times the t_i's. It equals a product of GL(m,C) characters at the n-th powers of those t_i's. The contribution is the proof method: substitute into the Weyl character formula and argue that the alternating sum cancels except for the desired terms, done combinatorially rather than through determinantal identities. That is the only new element. The abstract is clear about citing the 1934 and 2016 sources, so there is no claim of novelty on the statement itself. The combinatorial approach is presented as direct and independent, which is a reasonable goal if the cancellation pairing can be made explicit. The main soft spot is exactly the one the stress-test flags: whether the Weyl-group pairing really exhausts every non-contributing term for arbitrary m and n without missed fixed points or over-cancellation. The abstract does not show the bijection or length/inversion bookkeeping, so that step has to be checked in the body. If the pairing works cleanly it is a modest but honest alternative; if it leaves residuals the argument collapses. No circularity or invented entities appear in what is described. This is for readers who care about combinatorial proofs in the representation theory of GL(n) and might want to adapt the cancellation idea elsewhere. It is not urgent for the general literature. I would send it to referees because the claim is modest, the method is stated plainly, and a careful check of the pairing is feasible; it does not rise to the level of a desk reject. I would not cite it myself unless the cancellation turns out to be reusable.

Referee Report

1 major / 1 minor

Summary. The manuscript re-proves a theorem of Littlewood-Richardson and D. Prasad: for m,n ≥ 2 the character of an irreducible representation of GL(mn,C) evaluated at diagonal matrices with eigenvalues ω_n^{j-1} t_i (1≤i≤m, 1≤j≤n) equals a product of characters of GL(m,C) evaluated at diag(t_1^n,…,t_m^n). The new argument substitutes the indicated eigenvalues into the Weyl character formula and claims that all but the desired terms cancel via a direct combinatorial pairing on the Weyl group of GL(mn).

Significance. The result itself is classical, but the paper supplies an alternative to the usual determinantal identities by means of an explicit Weyl-group cancellation. If the pairing is shown to be exhaustive and free of residual fixed points, the approach may be useful for analogous factorizations in other classical groups where determinantal methods are unavailable.

major comments (1)

[main proof] The central combinatorial argument (the pairing that cancels non-contributing Weyl-group elements) is described only at the level of the abstract; no explicit bijection, length-function calculation, or verification that every non-fixed element is paired exactly once appears in the provided text. Without this, it is impossible to confirm that the cancellation exhausts all extraneous terms for arbitrary m,n.

minor comments (1)

[abstract/introduction] The abstract cites Theorem 2 of Prasad (Israel J. Math. 2016) and Theorem VI of Littlewood-Richardson (Q. J. Math. 1934); the introduction should state the precise statement being re-proved and note any minor differences in indexing or normalization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for greater explicitness in the combinatorial argument. We address the major comment below and will revise the paper accordingly.

read point-by-point responses

Referee: [main proof] The central combinatorial argument (the pairing that cancels non-contributing Weyl-group elements) is described only at the level of the abstract; no explicit bijection, length-function calculation, or verification that every non-fixed element is paired exactly once appears in the provided text. Without this, it is impossible to confirm that the cancellation exhausts all extraneous terms for arbitrary m,n.

Authors: We agree that the current manuscript presents the pairing at too high a level of generality. In the revised version we will supply an explicit, constructive bijection on the relevant subset of the Weyl group of GL(mn), together with a direct computation of the length function on paired elements (showing that each pair differs by an odd length and therefore contributes opposite signs) and a verification that the pairing is exhaustive, fixed-point-free on the non-contributing terms, and works uniformly for all m,n ≥ 2. These additions will make the cancellation argument fully rigorous and checkable. revision: yes

Circularity Check

0 steps flagged

No circularity: direct combinatorial proof independent of inputs

full rationale

The paper supplies an independent combinatorial argument via explicit cancellation in the Weyl group of GL(mn) to recover the known character factorization, replacing prior determinantal methods. No parameter fitting, self-definitional relations, or load-bearing self-citations appear; the cited results (Prasad 2016, Littlewood-Richardson 1934) are external classical theorems whose statements are not presupposed in the new pairing. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the standard Weyl character formula and properties of the Weyl group of GL(n,C); no free parameters or invented entities are introduced.

axioms (2)

standard math Weyl character formula expresses irreducible characters of GL(n,C)
Invoked to set up the character expression before cancellation.
standard math Weyl group acts on the set of weights with known sign and orbit structure
Central to the combinatorial cancellation argument.

pith-pipeline@v0.9.0 · 5692 in / 1186 out tokens · 23900 ms · 2026-05-24T10:50:35.699842+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We sum the Weyl numerator over various left cosets of a particularly chosen subgroup R which is isomorphic to Sn^m. We find that there is a subgroup C such that those cosets of R which have a representative from C contribute a nonzero term whereas sum over cosets not contained in CR contribute zero.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Aρmn(t·cn) = (−1)^{mn(m−1)(n−1)/4} ∏_{i<j}(ω^i_n − ω^j_n)^m × ∏_{i<j}(t^n_i − t^n_j)^n × (∏_{s=1}^m t_s)^{n(n−1)/2}

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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