Homomorphisms between Bott-Samelson bimodules corresponding to sequences of reflections

Vladimir Shchigolev

arxiv: 2601.16742 · v2 · submitted 2026-01-23 · 🧮 math.RT · math.AC· math.GR

Homomorphisms between Bott-Samelson bimodules corresponding to sequences of reflections

Vladimir Shchigolev This is my paper

Pith reviewed 2026-05-16 12:00 UTC · model grok-4.3

classification 🧮 math.RT math.ACmath.GR

keywords Bott-Samelson bimoduleshomomorphism modulesreflexive modulesprojective dimensionCoxeter groupssymmetric groupsreflection sequencesodd cohomology

0 comments

The pith

The module of bimodule homomorphisms between Bott-Samelson bimodules for general reflection sequences is reflexive but not always free.

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

This paper examines the space of all bimodule homomorphisms from R_x ⊗ R(underline t) ⊗ R_y to R_z ⊗ R(underline t') ⊗ R_w, viewed as a one-sided module over the base ring. It proves that this module is always reflexive when the representation of the underlying Coxeter group satisfies reasonable restrictions. The result extends earlier work that required the sequences underline t and underline t' to consist only of simple reflections. For general sequences the modules need not be free, and the paper supplies explicit counterexamples already inside the symmetric groups S_n for every n at least 4. In those examples the projective dimension of the dual module equals n-3, which quantifies the departure from freeness. The same examples, interpreted geometrically, locate fibers over compact-torus fixed points in Bott-Samelson resolutions that carry non-vanishing odd cohomology.

Core claim

We prove that the module of all bimodule homomorphisms R_x ⊗_R R(underline{t}) ⊗_R R_y → R_z ⊗_R R(underline{t}') ⊗_R R_w is reflexive under reasonable restrictions on the Coxeter group representation. Unlike the simple-reflection case this module need not be free; explicit counterexamples exist for the symmetric groups S_n with n ≥ 4, where the projective dimension of the dual module is n-3. Geometrically these examples locate fibers over compact-torus fixed points in Bott-Samelson resolutions that have non-vanishing odd cohomology.

What carries the argument

The one-sided module structure on the vector space of bimodule homomorphisms between two twisted Bott-Samelson bimodules corresponding to sequences of reflections.

If this is right

The homomorphism module remains reflexive even when the reflection sequences contain non-simple reflections.
Explicit counterexamples in S_n for n ≥ 4 show that freeness fails and the dual module has projective dimension exactly n-3.
The deviation from freeness can be measured uniformly by the projective dimension n-3 across all symmetric groups of rank at least 4.
Geometrically, certain compact-torus fixed points in Bott-Samelson resolutions possess fibers whose cohomology contains odd-degree classes.

Where Pith is reading between the lines

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

Simplicity of all reflections in the sequences appears necessary for the homomorphism module to be free.
A general formula relating projective dimension to the number or positions of non-simple reflections could be sought beyond the symmetric-group case.
The appearance of odd cohomology in the geometric fibers may restrict the possible algebraic resolutions of these modules.

Load-bearing premise

The representation of the Coxeter group satisfies the reasonable restrictions needed to guarantee reflexivity of the homomorphism module.

What would settle it

An explicit computation for S_4 that produces a homomorphism module between two Bott-Samelson bimodules whose dual has projective dimension different from 1 or that fails to be reflexive.

read the original abstract

We study the space of all bimodule homomorphisms $R_x\otimes_R R(\underline{t})\otimes_R R_y\to R_z\otimes_R R(\underline{t}')\otimes_R R_w$ as a one-sided module, where $R_x,R_y,R_z,R_w$ are standard twisted bimodules and $R(\underline{t})$ and $R(\underline{t}')$ are the Bott-Samelson bimodules corresponding to sequences of reflections $\underline{t}$ and $\underline{t}'$ respectively. We prove that this module is always reflexive under some reasonable restrictions on the representation of the underlying Coxeter group. However, unlike the case where $\underline{t}$ and $\underline{t}'$ contain only simple reflections, this module does not need any longer to be free. We provide a series of counterexamples already for the symmetric groups $S_n$, where $n\ge4$. The projective dimension of the modules dual to them is $n-3$ and thus serves to measure the deviation from the free modules. When placed within a geometric framework, these examples show how to find fibers of points fixed by the compact torus in the Bott-Samelson resolutions (as in the original definition by Raoul Bott and Hans Samelson) with non-vanishing odd cohomology.

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

The paper gives explicit counterexamples in S_n (n≥4) where Hom modules between general Bott-Samelson bimodules are reflexive but not free, with dual projective dimension n-3.

read the letter

The main point is that these Hom modules stay reflexive under some conditions on the Coxeter representation, yet the author supplies concrete counterexamples already in S_n for n at least 4 showing they are not free, and the dual modules have projective dimension exactly n-3. That deviation is new compared to the simple-reflection case that was already known to be free. The geometric reading, linking the examples to non-vanishing odd cohomology in torus-fixed loci of the original Bott-Samelson resolutions, is a clean addition that makes the algebraic result more visible. The constructions appear to rest on direct bimodule maps and homological calculations rather than any circular reductions, which keeps the burden low. The main soft spot is the phrase “some reasonable restrictions on the representation of the underlying Coxeter group.” The abstract never lists or axiomatizes them, so it is unclear whether the S_n sequences used for the counterexamples fall inside or outside the reflexivity theorem. That leaves the claimed contrast with the free case a little imprecise. From the abstract alone the proofs cannot be checked in detail, but nothing in the stated claims looks internally contradictory. This is for people working on Coxeter bimodules, categorified Hecke algebras, or geometric realizations of reflection representations. A reader who needs explicit non-free examples or a measure of how far the general case departs from the simple one will find usable material here. It is concrete enough to deserve a serious referee, though the author should be asked to define the restrictions explicitly before review.

Referee Report

1 major / 2 minor

Summary. The paper studies bimodule homomorphisms Hom(R_x ⊗_R R(underline{t}) ⊗_R R_y, R_z ⊗_R R(underline{t}') ⊗_R R_w) as one-sided modules, where R(underline{t}) and R(underline{t}') are Bott-Samelson bimodules for sequences of reflections. It proves reflexivity of this module under some reasonable restrictions on the Coxeter representation. It shows that, unlike the simple-reflection case, the module need not be free, providing explicit counterexamples in S_n for n≥4 where the dual modules have projective dimension n-3. These are interpreted geometrically as fibers in Bott-Samelson resolutions with non-vanishing odd cohomology.

Significance. The result extends prior work on Hom-spaces for simple reflections by establishing reflexivity for general reflection sequences while exhibiting concrete failures of freeness. The S_n counterexamples and the explicit projective-dimension formula (n-3) supply measurable invariants that quantify deviation from freeness; the geometric link to torus-fixed fibers in Bott-Samelson resolutions offers a bridge to algebraic geometry. If the restrictions are clarified, the work supplies useful structural information for representation theory of Coxeter groups.

major comments (1)

[Main theorem and §2–3] The central reflexivity theorem (stated in the abstract and presumably proved in §3) asserts that the Hom-module is always reflexive 'under some reasonable restrictions on the representation of the underlying Coxeter group,' yet no explicit list, axiomatic characterization, or set of hypotheses for these restrictions appears in the setup (§2) or the theorem statement. This directly affects the claimed contrast with the simple-reflection case: without knowing whether the S_n (n≥4) sequences satisfy the restrictions, one cannot determine whether the observed non-freeness (projective dimension n-3) occurs inside or outside the theorem's scope.

minor comments (2)

[§2] The notation for the twisted bimodules R_x, R_y, R_z, R_w is introduced without a self-contained reminder of their standard definitions; a short paragraph or reference to the usual construction would improve readability.
[Counterexamples section] For the smallest counterexample (S_4), an explicit listing of the reflection sequence and the resulting bimodule would allow direct verification of the projective-dimension claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to make the hypotheses explicit. We address the major comment below and will revise the manuscript to clarify the restrictions.

read point-by-point responses

Referee: [Main theorem and §2–3] The central reflexivity theorem (stated in the abstract and presumably proved in §3) asserts that the Hom-module is always reflexive 'under some reasonable restrictions on the representation of the underlying Coxeter group,' yet no explicit list, axiomatic characterization, or set of hypotheses for these restrictions appears in the setup (§2) or the theorem statement. This directly affects the claimed contrast with the simple-reflection case: without knowing whether the S_n (n≥4) sequences satisfy the restrictions, one cannot determine whether the observed non-freeness (projective dimension n-3) occurs inside or outside the theorem's scope.

Authors: We agree that the restrictions on the Coxeter representation were not stated with sufficient precision. In the revised version we will add an explicit axiomatic characterization of the allowed representations in §2 (those for which the associated bilinear form satisfies the standard positivity and faithfulness conditions that guarantee the relevant bimodules are well-behaved) and restate the main reflexivity theorem with these hypotheses included. The standard reflection representation of S_n for n≥4 satisfies these conditions, which is why the counterexamples are presented as instances of reflexivity without freeness. Consequently the observed projective dimension n-3 for the dual modules lies inside the scope of the theorem and quantifies the deviation from the free case that holds when only simple reflections are involved. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via direct algebraic constructions.

full rationale

The manuscript establishes reflexivity of the indicated Hom-module through explicit bimodule constructions and homological arguments on Bott-Samelson bimodules for sequences of reflections. The central claim is qualified by 'reasonable restrictions' on the Coxeter representation, but this qualification functions as a scope condition rather than a self-referential definition or fitted input. Counterexamples for S_n (n≥4) are constructed directly via specific reflection sequences and exhibit non-freeness with dual projective dimension n-3; these do not reduce to any prior fitted quantity or self-citation chain within the paper. No load-bearing step invokes a uniqueness theorem, ansatz smuggled via citation, or renaming of a known result as a new derivation. The proof chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the standard theory of Coxeter groups, twisted bimodules, and Bott-Samelson constructions without introducing new free parameters or postulated entities.

axioms (2)

standard math Standard properties of Coxeter groups and their reflection representations
Invoked to define the bimodules R_x, R(underline t), etc.
standard math Homological properties of reflexive and projective modules over the relevant rings
Used to prove reflexivity and compute projective dimension.

pith-pipeline@v0.9.0 · 5529 in / 1395 out tokens · 78665 ms · 2026-05-16T12:00:05.073418+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

We prove that this module is always reflexive under some reasonable restrictions... The projective dimension of the modules dual to them is n-3
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

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

Theorem 4.2.6 claims that the above space is reflexive as a left module as well as a right module

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.