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arxiv: 2507.04460 · v3 · submitted 2025-07-06 · 🧮 math.RT

Homological dimensions of Schur algebras S(p,2p) and an Auslander-type correspondence

Tiago Cruz , Karin Erdmann This is my paper

Pith reviewed 2026-05-19 05:57 UTC · model grok-4.3

classification 🧮 math.RT
keywords Schur algebrasSchur-Weyl dualityAuslander correspondencerelative Auslander pairhomological dimensionsYoung modulestilting modulessymmetric groups
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The pith

Schur-Weyl duality for Schur algebras S(p, 2p) is realized as an Auslander-type correspondence.

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

This paper establishes that Schur-Weyl duality between the Schur algebra S(p, 2p) and the centralizer algebra Λ(p, 2p) is an instance of an Auslander-type correspondence. It computes the global dimension of S(p, 2p) and the relative dominant dimension with respect to the tensor space (k^p)⊗2p. The work shows that this pair forms a relative 4(p-1)-Auslander pair and determines the Hemmer-Nakano dimension for the quasi-hereditary cover of Λ(p, 2p). It also shows that a sum of certain Young modules is a full tilting module when p>2. A reader would care because this links classical duality to modern homological algebra in the context of representations of symmetric groups and algebraic groups in positive characteristic.

Core claim

We prove that Schur-Weyl duality between S(p, 2p) and Λ(p, 2p) is an instance of an Auslander-type correspondence. We show that the pair (S(p, 2p), (k^p)⊗2p) forms a relative 4(p-1)-Auslander pair in the sense of Cruz and Psaroudakis. We compute the global dimension and relative dominant dimension, determine the Hemmer-Nakano dimension, and show that the direct sum of some Young modules over Λ(p, 2p) is a full tilting module when p>2.

What carries the argument

The relative 4(p-1)-Auslander pair (S(p, 2p), (k^p)⊗2p) that realizes the Schur-Weyl duality as an Auslander correspondence.

If this is right

  • The global dimension of the Schur algebra S(p, 2p) follows from its relative dominant dimension with respect to the tensor space.
  • The Hemmer-Nakano dimension is determined for the quasi-hereditary cover arising from the duality.
  • A direct sum of Young modules over Λ(p, 2p) forms a full tilting module for p greater than 2.
  • The connection allows viewing Schur-Weyl duality through the lens of higher homological algebra.

Where Pith is reading between the lines

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

  • If this holds for S(p, 2p), analogous results may exist for Schur algebras with other parameters relating tensor degree to the characteristic.
  • The Auslander correspondence here might extend to other centralizer algebras in group representation theory.
  • Computations of these dimensions could help classify tilting modules in modular representation theory of symmetric groups.

Load-bearing premise

The assumption that Λ(p, 2p) is precisely the centraliser algebra of the tensor space (k^p)⊗2p as a module over S(p, 2p).

What would settle it

An explicit computation for a small prime p, such as p=3, showing that the relative dominant dimension with respect to (k^p)⊗2p is not 4(p-1).

read the original abstract

We study the homological properties of Schur algebras $S(p, 2p)$ over a field $k$ of positive characteristic $p$, focusing on their interplay with the representation theory of quotients of group algebras of symmetric groups via Schur-Weyl duality. Schur-Weyl duality establishes that the centraliser algebra, $\Lambda(p, 2p)$, of the tensor space $(k^p)^{\otimes 2p}$ (as a module over $S(p, 2p)$) is a quotient of the group algebra of the symmetric group. In this paper, we prove that Schur-Weyl duality between $S(p, 2p)$ and $\Lambda(p, 2p)$ is an instance of an Auslander-type correspondence. We compute the global dimension of Schur algebras $S(p, 2p)$ and their relative dominant dimension with respect to the tensor space $(k^p)^{\otimes 2p}$. In particular, we show that the pair $(S(p, 2p), (k^p)^{\otimes 2p})$ forms a relative $4(p-1)$-Auslander pair in the sense of Cruz and Psaroudakis, thereby connecting Schur algebras with higher homological algebra. Moreover, we determine the Hemmer-Nakano dimension associated with the quasi-hereditary cover of $\Lambda(p, 2p)$ that arises from Schur-Weyl duality. As an application, we show that the direct sum of some Young modules over $\Lambda(p, 2p)$ is a full tilting module when $p>2$.

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.

Referee Report

1 major / 2 minor

Summary. The paper examines homological properties of the Schur algebra S(p, 2p) in characteristic p, establishing that Schur-Weyl duality with the centraliser algebra Λ(p, 2p) of the tensor space (k^p)^⊗2p realises an Auslander-type correspondence in the sense of Cruz and Psaroudakis. It computes the global dimension of S(p, 2p), determines the relative dominant dimension of S(p, 2p) with respect to (k^p)^⊗2p, proves that the pair (S(p, 2p), (k^p)^⊗2p) is a relative 4(p-1)-Auslander pair, identifies the associated Hemmer-Nakano dimension for the quasi-hereditary cover of Λ(p, 2p), and shows that a direct sum of certain Young modules over Λ(p, 2p) is a full tilting module when p > 2.

Significance. If the stated bounds and correspondence hold, the work supplies explicit homological invariants for a family of Schur algebras that arise in the representation theory of symmetric groups, thereby linking classical Schur-Weyl duality to higher Auslander-Reiten theory. The explicit tilting-module application and the determination of the Hemmer-Nakano dimension constitute concrete, usable results for the study of quasi-hereditary covers.

major comments (1)
  1. [Main theorem on the relative 4(p-1)-Auslander pair (abstract and § on relative dominant dimension)] The central claim that (S(p, 2p), (k^p)^⊗2p) forms a relative 4(p-1)-Auslander pair rests on showing that the relative dominant dimension is at least 4(p-1). In characteristic p the tensor space decomposes into Young modules whose minimal relative projective resolutions involve Specht filtrations; any non-vanishing relative Ext groups in degrees near 2(p-1) would force the relative dominant dimension to be strictly smaller than 4(p-1) and thereby invalidate the exact Auslander-pair statement. The manuscript must therefore exhibit either an explicit computation of the relative Ext groups up to degree 4(p-1) or a vanishing argument that accounts for the characteristic-p behaviour of the Specht filtration.
minor comments (2)
  1. [Notation and preliminaries] Clarify the precise definition of the relative dominant dimension used (with respect to the tensor-space module) and confirm that it coincides with the definition in Cruz-Psaroudakis.
  2. [Application section] Add a short table or explicit list of the Young modules whose direct sum is claimed to be tilting, together with the range of p for which the statement holds.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for raising this important point about the proof of the relative dominant dimension. We address the concern directly below.

read point-by-point responses

  1. Referee: [Main theorem on the relative 4(p-1)-Auslander pair (abstract and § on relative dominant dimension)] The central claim that (S(p, 2p), (k^p)^⊗2p) forms a relative 4(p-1)-Auslander pair rests on showing that the relative dominant dimension is at least 4(p-1). In characteristic p the tensor space decomposes into Young modules whose minimal relative projective resolutions involve Specht filtrations; any non-vanishing relative Ext groups in degrees near 2(p-1) would force the relative dominant dimension to be strictly smaller than 4(p-1) and thereby invalidate the exact Auslander-pair statement. The manuscript must therefore exhibit either an explicit computation of the relative Ext groups up to degree 4(p-1) or a vanishing argument that accounts for the characteristic-p behaviour of the Specht filtration.

    Authors: We thank the referee for this observation. In the section on relative dominant dimension we do supply a vanishing argument for the relative Ext groups that explicitly incorporates the characteristic-p behaviour of the Specht filtrations. The argument proceeds by first decomposing the tensor space into a direct sum of Young modules, then using the known Specht filtrations of these modules together with the weight-space decomposition for S(p,2p). For the weights appearing in (k^p)^⊗2p when the second parameter is 2p, the higher Ext groups between the relevant modules vanish in degrees 1 through 4(p-1)-1 because the Specht modules in this range are either projective or admit filtrations whose successive quotients have no non-zero extensions below the stated bound (this follows from the explicit description of the radical layers and the fact that p divides the relevant hook lengths only in controlled ways). The same vanishing is used to establish that the relative dominant dimension is at least 4(p-1), confirming the Auslander-pair statement. We are happy to expand the intermediate steps of this calculation if the referee finds the current presentation insufficiently detailed. revision: partial

Circularity Check

1 steps flagged

Minor self-citation to definition of relative Auslander pair; central homological computations remain independent

specific steps
  1. self citation load bearing [Abstract]
    "we show that the pair (S(p, 2p), (k^p)⊗2p) forms a relative 4(p-1)-Auslander pair in the sense of Cruz and Psaroudakis"

    The central claim adopts the technical definition and framework directly from a citation whose lead author overlaps with the present paper; while this does not force the numerical bound or the duality instance by construction, it qualifies as a minor self-citation that frames the result.

full rationale

The paper invokes the definition of a relative Auslander pair from prior work by Cruz and Psaroudakis to frame its main result, but this is a standard definitional reference rather than a load-bearing reduction. The computations of global dimension and relative dominant dimension for S(p, 2p) with respect to the tensor space rely on standard Schur-Weyl duality and homological algebra techniques applied to the specific case, without the target claims reducing by construction to fitted inputs or self-referential equations. No self-definitional loops, fitted predictions, or ansatz smuggling are exhibited in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard domain assumption of Schur-Weyl duality in positive characteristic together with background results from homological algebra; no free parameters or newly invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Schur-Weyl duality identifies the centraliser algebra Λ(p,2p) of the tensor space (k^p)⊗2p as a quotient of the symmetric group algebra
    This identification is the starting point used to view the duality as an Auslander-type correspondence.

pith-pipeline@v0.9.0 · 5831 in / 1521 out tokens · 34724 ms · 2026-05-19T05:57:37.354650+00:00 · methodology

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Works this paper leans on

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