Semisimplifying categorical Heisenberg actions and periodic equivalences

Alexander Sherman; Bregje Pauwels; Chris Hone; Finn Klein; Oded Yacobi; Victor L. Zhang

REVIEW 3 major objections 5 minor 40 references

Semisimplifying categorical Heisenberg actions and periodic equivalences

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A kernel-and-quotient recipe builds global functors that restrict to the periodic equivalences for symmetric group representations and commute with the categorical affine sl_p action.

desk verdict Explicit global functors for the periodic equivalences, with real content; the main soft spot is a one-line support-theory citation. read the letter →

arxiv 2509.07377 v2 pith:WBDEMLQV submitted 2025-09-09 math.RT

Chris Hone , Finn Klein , Bregje Pauwels , Alexander Sherman , Oded Yacobi , Victor L. Zhang This is my paper · ORCID

classification math.RT MSC 20C3020C2018M0517B67

keywords semisimplificationOTIfunctorcategoricalHeisenbergactionsymmetricgroupsperiodicequivalencespolynomialfunctorsVerlindecategorymodularrepresentationtheory

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 constructs a family of explicit functors, called OTI functors, by restricting to a cyclic p-subgroup and then semisimplifying. For symmetric groups, two concrete quotient functors by kernels of a generic nilpotent operator z_a extend the known periodic equivalences between Rep_k S_n and Rep_k S_{n-p^r} from subcategories to the whole category. These functors commute with the categorical Heisenberg action and, on mod-p Grothendieck groups, induce the central endomorphism e^p. The same machinery categorifies the canonical fixed-point correspondence between H-fixed simple modules and simples of the fixed-point subgroup, and it yields a branching statement: stable objects restrict to a p-subgroup as a trivial module plus indecomposable summands of p-divisible dimension.

What carries the argument

The OTI functor: compose the p^r-fold Heisenberg functor E^{p^r} (explicitly, restriction to S_{n-p^r} × S_{p^r}) with a CF functor φ: Rep_k H → A, where CF means φ vanishes on induction from any non-transitive subgroup. For symmetric groups, taking H = A ≅ (C_p)^r and φ = φ_a attached to the shifted cyclic operator z_a = a_1(σ_1−1)+⋯+a_r(σ_r−1) yields the kernel-quotient functors of Theorem A; the CF condition makes the construction see only the transitive part of the action. Semisimplification into the Verlinde category supplies the monoidal examples, and the diagrammatic presentation of the Heisenberg category is used to prove the morphism-of-actions property.

What would settle it

Take p=5, n=11, and the stable partition λ=(8,3). With any generic a for the transitive subgroup A=C_5, compute the OTI functor Φ_a on the Specht module S^{(8,3)} and check whether the result is isomorphic to S^{(3,3)} in Rep S_6. The theorem predicts this isomorphism; a single non-isomorphism would falsify the periodic-equivalence claim.

Watch

Extended reading notes

Core claim

The central claim is that global semisimplification functors — restriction to a p^r-cycle subgroup followed by a CF functor φ — define morphisms of degenerate categorical Heisenberg actions, and that for the specific φ coming from a generic shifted cyclic subgroup z_a they restrict to the periodic equivalences on polynomial functors and symmetric group representations. In particular, for any generic tuple a, the functors M ↦ ker(z_a)/(ker(z_a) ∩ im(z_a)) and M ↦ ker(z_a)/im(z_a^{p-1}) send Rep_k S_n to Rep_k S_{n-p^r}, agree with removing p^r boxes from the first row on stable partitions, and intertwine the categorical action of the affine Lie algebra sl_p. A further consequence is that ever

Load-bearing premise

The load-bearing premise is a genericity assumption on the tuple a: the shifted nilpotent operator z_a is not supported on any proper subgroup of the elementary abelian p-subgroup A, so every module induced from a proper subgroup is free over z_a; if this failed, the quotient functors would not realize the periodic equivalences.

Editorial extensions

If this is right

The periodic equivalences of [27] and [24] are realized by global, explicitly defined functors on all of Rep_k S_n, not merely on stable subcategories.
Every generic tuple a gives such a functor, and two explicit quotient formulas always work, so the equivalences are part of a family rather than isolated constructions.
Because the functors commute with the categorical Heisenberg action, they induce endomorphisms of the mod-p basic representation of affine sl_p by the central element e^p; this reproves reducibility of that representation with an infinite composition series.
The canonical fixed-point correspondence is categorified: the extension of an H-fixed simple module maps to its correspondent in the appropriate Verlinde component, explaining the sign in the correspondence.
Objects in the stable subcategory S_{n,p^r} restrict to H as a trivial module plus indecomposable summands of dimension divisible by p.

Reading between the lines

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

The same OTI construction should yield global equivalences for other categorical actions, including central charge zero actions such as the GL_n tilting category, where the paper notes a Donkin tensor-product analogue; this is a conjecture, not proved here.
Because the functors are defined by kernel-and-quotient formulas, they are directly computable for small n; one could tabulate their effect on Specht modules and decomposition numbers to probe behavior beyond the stable range.
The CF condition is a minimal support-theoretic hypothesis; relaxing it so that non-transitive contributions are merely negligible rather than zero might produce additional functors that still commute with the action and still induce e^p on Grothendieck groups.
The Verlinde-component interpretation of the sign in the fixed-point correspondence suggests that other correspondences with sign ambiguities may admit similar categorical explanations.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a strong paper and the reader's ACCEPT is about right. The new thing is explicit, global functors Rep(S_n) -> Rep(S_{n-p^r}) that restrict to Henke-Koenig/Harman periodic equivalences and commute with the categorical csl_p action. Prior work only had inexplicit equivalences on subcategories; here you get computable kernels of z_a modulo image pieces, plus a symmetric monoidal globalization via polynomial functors. The OTI functor framework is genuinely useful: it packages the construction, gives a categorical Glauberman correspondence with the sign explained, and Theorem B is proven with real diagrammatic work. None of this looks circular; the periodic equivalences are prior results, and the paper is a new realization plus compatibility.

I share the reader's instinct that the softest load-bearing point is Lemma 8.3.2. It asserts phi_a is a CF functor by saying the support-theoretic containment follows from [19, Prop. 8.2.4]. That's a one-sentence proof for exactly the property that makes the explicit functors in Theorem A restrict to the periodic equivalences. I don't see a falsehood: for r=2 the claim matches direct rank-variety checks, and the general statement is plausible. But the paper should either state the precise content of Prop 8.2.4 or give a self-contained argument. As written, a referee needs to do that verification. That's a revision request, not a rejection.

Other notes: the paper is long and the diagrammatic Section 9 is a lot, but the core logic is coherent. Some proofs leave routine checks to the reader, and Remark 2.2.3 honestly records a limitation that doesn't affect the main results. Citation pattern looks fair: the periodic equivalences are credited to [24,27], OTI traces to [2] (Baine et al.) and [10] for V-functors. No fitting red flags. The 'generic tuple a' is a genuine parameter, not a tuned constant.

Bottom line: this deserves a serious referee. The main theorem is important enough and the proof is substantial. Ask the authors to nail down Lemma 8.3.2 and maybe shorten the diagrammatics. I'd cite it if I were working in this area.

Referee Report

3 major / 5 minor

Summary. The paper introduces One Tree Island (OTI) functors, defined by composing a categorical action of E^{p^r} with a CF functor on representations of a transitive subgroup of S_{p^r}, and studies them in modular representation theory and degenerate categorical Heisenberg actions. The main results are: (1) an OTI functor categorifies the cyclic Glauberman correspondence (Theorem 4.5.3); (2) every OTI functor attached to a CF functor is a morphism of degenerate categorical Heisenberg actions, and the semisimplification case induces the central element e^p on mod p Grothendieck groups (Theorem B, Sections 5.6–5.7 and 9); (3) for a generic shifted cyclic subgroup, the associated OTI functors give explicit global functors Rep_k S_n → Rep_k S_{n-p^r} that restrict to the Henke–Koenig periodic equivalences and commute with the csl_p categorical action (Theorem A, Theorem 8.3.3, Theorem 8.5.3); (4) analogous equivalences hold for strict polynomial functors, with applications to branching rules for symmetric groups (Theorem C, Corollary 8.5.4). The proofs combine highest-weight-category machinery, Schur–Weyl duality, support-theoretic facts for elementary abelian p-groups, and a diagrammatic proof of compatibility with the categorical action.

Significance. If the main theorems are correct, the paper provides a genuinely new and explicit global realization of the periodic equivalences of Henke–Koenig, replacing inexplicit constructions that pass through Schur–Weyl duality and subcategory restrictions. The functors are concrete (kernels and images of generic nilpotent operators), symmetric monoidal in several cases, and compatible with the categorical csl_p action, which is a strong and useful feature. The paper also gives a clean categorical interpretation of the sign in the Glauberman correspondence and derives branching statements that are of independent interest. The proof strategy is well structured: the highest-weight-category reduction, the use of Lemma 3.3.3 for exactness of semisimplification, and the diagrammatic proof of Theorem 5.6.1 are appropriate. The authors are careful to distinguish their contributions from earlier work of Henke–Koenig, Martin–Woodcock, and Harman, and the central claims appear internally consistent. The main weaknesses are places where load-bearing steps are asserted rather than demonstrated, most notably the support-theoretic proof of the CF property for shifted cyclic subgroups.

major comments (3)

[Section 8.3, Lemma 8.3.2] This lemma is the only place where the crucial CF property of φ_a is established, and it is load-bearing for Theorem 8.3.3 and hence for Theorem A. The proof is a single sentence: “this follows from [19, Prop. 8.2.4].” The proposition is not stated, and the genericity condition on a is not connected to it. Please state the precise support-theoretic fact used, verify that it applies to Ind_A^{A'}(M) for every proper subgroup A'⊂A, and explain why it implies that Ind_A^{A'}(M) is projective as an R_a-module. Without this, the restriction of the explicit functors to the periodic equivalences is not fully established.
[Section 8.3, proof of Theorem 8.3.3] The proof asserts: “By induction on r, repeatedly applying Lemma 3.3.3 shows that Φ_a(Δ_λ)→Φ_a(T^•) is exact and that Φ_a(Δ_λ)→tilde Φ_a(Δ_λ) is an isomorphism.” Lemma 3.3.3 requires the relevant short exact sequences to split as R-modules; that splitting is exactly what needs to be checked for the terms of the tilting resolution after applying E^{p^r}. The present argument is too compressed to verify that the induction hypothesis applies at each step. Please provide the details of the induction or an alternative proof that the tilting resolution is split over R_a after applying E^{p^r}.
[Section 9, Proposition 9.4.5 and Theorem 5.6.1] The proof of Proposition 9.4.5 treats the case k≥0 and states that k<0 is similar. However, the main application to symmetric groups, Example 5.2.1, has central charge k=−1, so the k<0 case is the one needed for Theorem B(1) in the paper’s central example. Since the structural Lemma 9.4.2 and Proposition 9.4.5 both rely on the inverse of the appropriate isomorphism (9.2.3) or (9.2.4), the similarity is not completely formal. Please write out the k<0 argument or give a precise symmetry that reduces it to the k≥0 case.

minor comments (5)

[Section 8.3, notation] The tuple a is written as (a_1,...,a_n) with n in the display after “we will fix a=(a_1,a_2,...,a_n)∈k^n”, even though A≅C_p^r has r generators. It should be a∈k^r, as in the introduction and Theorem A.
[Section 8.5, definition of Rep^st_{pr}S_d] The sentence “define Rep^st_{pr} S_d ... to be the full subcategory of Pol^st_{d,pr} ... on objects in the image of F⊗sgn” appears garbled: Rep^st_{pr}S_d should be a subcategory of Rep_k S_d, not of Pol^st. Please correct the wording.
[Section 8.1] Typo: “recollement of abelian catetories” should be “categories”.
[Section 8.4, Corollary 8.5.4] In the statement “V|_H = 1^{⊕ℓ} ⊕ ⊕_i N_i”, the first 1 should be written as 1^{⊕ℓ} to avoid confusion with the trivial module of dimension one. This is a presentation issue only.
[Section 1.4 and Theorem C] The introduction states that the functors in Theorem A “are also examples of OTI functors,” but the verification that ker(z_a)/im(z_a^{p-1}) is a CF functor, or satisfies the hypotheses of Theorem 8.5.3, is not given explicitly. A short remark after Lemma 8.3.2 would help the reader connect Theorem A to the general machinery.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem A is an explicit global realization of the already-established Henke–Koenig periodic equivalences; load-bearing citations are external.

full rationale

Walking the derivation chain, the central claims do not reduce to their inputs. The paper's own framing is that the periodic equivalences S_{n,p^r} ≅ S_{n-p^r,p^r} were first established in [27] and [24]; Section 6 says "This result was essentially proven in [27] and [24], but we recast the proof in terms of OTI functors," and the abstract says the paper "globalize[s] the equivalences of Henke–Koenig." Thus Theorem A is a new explicit global functor realizing a known equivalence, not a prediction forced by a fitted parameter. The explicit functors ker(z_a)/(ker∩im z_a) and ker(z_a)/im(z_a^{p-1}) are defined from the action of a generic shifted cyclic subgroup z_a; a is any generic tuple with no fitting step, and the proof that φ_a is a CF functor (Lemma 8.3.2) is the only delicate load-bearing link. That proof is a one-sentence appeal to the external textbook [19, Prop. 8.2.4] ("However this follows from [19, Prop. 8.2.4]"), so the support-theoretic fact is borrowed, not re-derived; but it is not a self-citation and does not presuppose the target equivalence. A reader verifying the paper would want the content of [19, Prop. 8.2.4] spelled out, but that is a correctness/verification gap, not circularity. Theorem B(1) is a diagrammatic proof that all CF functors give morphisms of categorical actions; Theorem B(2)'s e^p statement is a direct Grothendieck-group telescoping computation (Proposition 5.7.1, Corollary 5.7.2), not a renaming of a known result. Self-citations (e.g. [2], overlapping with author Sherman) are motivational and are not the load-bearing argument. No equation is defined in terms of its conclusion; the score is 0.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The construction introduces no new entities. Free parameters are the generic tuple a, which is a genuine input of the construction but not fitted to the target result. Most axioms are standard results from tensor categories, highest weight categories, and modular representation theory that are explicitly cited. The only assumption specific to this paper's machinery is the CF condition and the support-theoretic fact used to prove it.

free parameters (1)

generic coefficient tuple a = any a_1,...,a_r in k linearly independent over F_p
Chosen to define the shifted cyclic subalgebra R_a; the results are independent of the choice, so this is not fitted to data.

assumptions (6)

standard math Semisimplification of Rep C_p equals the componentwise functor phi of (3.3.1) and is monoidal (Prop 3.3.2, from [18]).
Used throughout to define OTI functors and compute their Grothendieck-group effect; cited to Etingof-Ostrik.
standard math The Verlinde category Ver_p is a semisimple symmetric tensor category with the stated Grothendieck ring and K0(Ver_p) acting on F_p via dim_p (Section 3, [17,18]).
Provides the target category for semisimplification and the mod-p dimension function; external standard result.
domain assumption Degenerate categorical Heisenberg actions satisfy the relations of [7] and give an action of S_n on E^n, hence an action of C_p (Section 5.2).
Defines the categorical setting in which OTI functors are constructed; taken as input framework from Brundan-Savage-Webster.
domain assumption Pol_d is a highest weight category with the stated standard, costandard and tilting objects, and Lemma 7.1.3's equivalences from [20,29] hold.
Supports the highest-weight-category machinery used in Section 8; cited to Friedlander-Suslin and Hong-Yacobi.
domain assumption Support-theoretic characterization: for generic a, a avoids the support of induced modules from proper subgroups (Lemma 8.3.2 relies on [19, Prop. 8.2.4]).
This is the key input making phi_a a CF functor; if false, the Theorem A functors would lose the CF property.
standard math An equivalence on tilting modules extends to an equivalence of highest weight categories ([24, Cor. 1.6]).
Used in Theorem 8.3.3 to pass from tilting-level equivalences to full abelian equivalences; cited to Harman.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Semisimplifying categorical Heisenberg actions and periodic equivalences." pith.science (2026). https://pith.science/paper/WBDEMLQV

@misc{pith2026250907377,
  author       = {Pith},
  title        = {Pith review of: Semisimplifying categorical Heisenberg actions and periodic equivalences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WBDEMLQV}},
  note         = {Machine review of arXiv:2509.07377}
}

abstract

We systematically apply semisimplification functors in modular representation theory. Motivated by the Duflo--Serganova functor in Lie superalgebras, we construct various functors of interest. In the setting of finite groups, we refine the cyclic group Brauer construction and categorify the Glauberman correspondence. In the setting of degenerate categorical Heisenberg actions, we obtain a rich collection of functors which commute with the categorical action. Applied to well-known categorifications of the basic representation and Fock space, our functors give explicit realizations of periodic equivalences for polynomial functors and symmetric groups first studied by Henke-Koenig. This allows us to globalize the equivalences of Henke-Koenig by symmetric monoidal functors. We apply these results to deduce branching properties of certain modular representations of $S_n$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 38 canonical work pages

[2]

A remarkable functor on $G$-modules

J. Baine, T. Fell, A. Romanov, A. Sherman, and G. Williamson, A remarkable functor on G-modules. ArXiv preprint 2505.07144 (2025)

work page Pith review arXiv 2025
[10]

Towards higher Frobenius functors for symmetric tensor categories

Coulembier, Kevin, and Johannes Flake. ”Towards higher Frobenius functors for symmetric tensor categories.” arXiv preprint arXiv:2405.19506 (2024)

work page Pith review arXiv 2024
[1]

J. L. Alperin,Local representation theory, Cambridge Studies in Advanced Mathematics, 11, Cambridge Univ. Press, Cambridge, 1986; MR0860771

work page 1986
[3]

Balmer, The spectrum of prime ideals in tensor triangulated categories, J

P. Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math.588(2005), 149–168; MR2196732

work page 2005
[4]

J. W. Barrett and B. W. Westbury, Spherical categories, Adv. Math.143(1999), no. 2, 357– 375; MR1686423

work page 1999
[5]

D. J. Benson,Representations of elementary abelianp-groups and vector bundles, Cambridge Tracts in Mathematics, 208, Cambridge Univ. Press, Cambridge, 2017; MR3585474

work page 2017
[6]

Brundan, On the definition of Heisenberg category, Algebr

J. Brundan, On the definition of Heisenberg category, Algebr. Comb.1(2018), no. 4, 523–544; MR3875075

work page 2018
[7]

Brundan, A

J. Brundan, A. Savage and B. Webster, Heisenberg and Kac-Moody categorification, Selecta Math. (N.S.)26(2020), no. 5, Paper No. 74, 62 pp.; MR4169511

work page 2020

Show all 40 references

[8]

Brundan and C

J. Brundan and C. Stroppel, Semi-infinite highest weight categories, Mem. Amer. Math. Soc. 293(2024), no. 1459, vii+152 pp.; MR4684337

2024
[9]

Coulembier, P

K. Coulembier, P. I. Etingof and V. Ostrik, On Frobenius exact symmetric tensor categories, Ann. of Math. (2)197(2023), no. 3, 1235–1279; MR4564264

2023
[11]

De Concini, V

C. De Concini, V. G. Kac and D. A. Kazhdan, Boson-fermion correspondence overZ, in Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), 124–137, Adv. Ser. Math. Phys., 7, World Sci. Publ., Teaneck, NJ, ; MR1026951

1988
[12]

Deligne, Cat´ egories tannakiennes, inThe Grothendieck Festschrift, Vol

P. Deligne, Cat´ egories tannakiennes, inThe Grothendieck Festschrift, Vol. II, 111–195, Progr. Math., 87, Birkh¨ auser Boston, Boston, MA, ; MR1106898
[13]

Donkin, On tilting modules for algebraic groups, Math

S. Donkin, On tilting modules for algebraic groups, Math. Z.212(1993), no. 1, 39–60; MR1200163 39

1993
[14]

”On associated variety for Lie superalgebras.” arXiv preprint math/0507198 (2005)

Duflo, Michel, and Vera Serganova. ”On associated variety for Lie superalgebras.” arXiv preprint math/0507198 (2005)

2005 arXiv
[15]

Entova-Aizenbud, V

I. Entova-Aizenbud, V. Hinich and V. V. Serganova, Deligne categories and the limit of cate- gories Rep(GL(m|n)), Int. Math. Res. Not. IMRN2020, no. 15, 4602–4666; MR4130848
[16]

P. I. Etingof, S. Gelaki, D. Nikshych and V. Ostrik,Tensor categories, Mathematical Surveys and Monographs, 205, Amer. Math. Soc., Providence, RI, 2015; MR3242743

2015
[17]

P. I. Etingof and V. Ostrik, On the Frobenius functor for symmetric tensor categories in positive characteristic, J. Reine Angew. Math.773(2021), 165–198; MR4237968

2021
[18]

P. I. Etingof and V. Ostrik, On semisimplification of tensor categories, inRepresentation theory and algebraic geometry—a conference celebrating the birthdays of Sasha Beilinson and Victor Ginzburg, 3–35, Trends Math., Birkh¨ auser/Springer, Cham, ; MR4486913
[19]

Evens,The cohomology of groups, Oxford Mathematical Monographs Oxford Science Pub- lications, , Oxford Univ

L. Evens,The cohomology of groups, Oxford Mathematical Monographs Oxford Science Pub- lications, , Oxford Univ. Press, New York, 1991; MR1144017

1991
[20]

E. M. Friedlander and A. A. Suslin, Cohomology of finite group schemes over a field, Invent. Math.127(1997), no. 2, 209–270; MR1427618

1997
[21]

Glauberman, Correspondences of characters for relatively prime operator groups, Canadian J

G. Glauberman, Correspondences of characters for relatively prime operator groups, Canadian J. Math.20(1968), 1465–1488; MR0232866

1968
[22]

Gorelik and T

M. Gorelik and T. Heidersdorf, Gruson-Serganova character formulas and the Duflo-Serganova cohomology functor, J. Reine Angew. Math.798(2023), 1–54; MR4579700

2023
[23]

Gorelik, C

M. Gorelik, C. Hoyt, V. Serganova, and A. Sherman, The Duflo–Serganova functor, vingt ans apr´ es, Journal of the Indian Institute of Science 102.3 (2022): 961-1000

2022
[24]

”Stability and periodicity in the modular representation theory of symmetric groups.” arXiv preprint arXiv:1509.06414 (2015)

Harman, Nate. ”Stability and periodicity in the modular representation theory of symmetric groups.” arXiv preprint arXiv:1509.06414 (2015)

2015 arXiv
[25]

Heidersdorf and R

T. Heidersdorf and R. Weissauer, Cohomological tensor functors on representations of the gen- eral linear supergroup, Mem. Amer. Math. Soc.270(2021), no. 1320, v+106 pp.; MR4276967

2021
[26]

D. J. Hemmer, Symmetric group modules with Specht and dual Specht filtrations, Comm. Algebra35(2007), no. 11, 3292–3306; MR2362656

2007
[27]

A. E. Henke and S. Koenig, Relating polynomial GL(n)-representations in different degrees, J. Reine Angew. Math.551(2002), 219–235; MR1932179

2002
[28]

J. Hong, A. Touz´ e and O. Yacobi, Polynomial functors and categorifications of Fock space, inSymmetry: representation theory and its applications, 327–352, Progr. Math., 257, Birkh¨ auser/Springer, New York, ; MR3363015
[29]

Hong and O

J. Hong and O. Yacobi, Polynomial representations and categorifications of Fock space, Algebr. Represent. Theory16(2013), no. 5, 1273–1311; MR3102954

2013
[30]

Hong and O

J. Hong and O. Yacobi, Polynomial functors and categorifications of Fock space II, Adv. Math. 237(2013), 360–403; MR3028582

2013
[31]

C. F. Hoyt and S. Reif, Grothendieck rings for Lie superalgebras and the Duflo-Serganova functor, Algebra Number Theory12(2018), no. 9, 2167–2184; MR3894432

2018
[32]

G. D. James,The representation theory of the symmetric groups, Lecture Notes in Mathematics, 682, Springer, Berlin, 1978; MR0513828

1978
[33]

G. D. James, The decomposition matrices of GL n(q) forn≤10, Proc. London Math. Soc. (3) 60(1990), no. 2, 225–265; MR1031453

1990
[34]

A. S. Kleshchev, Branching rules for modular representations of symmetric groups. II, J. Reine Angew. Math.459(1995), 163–212; MR1319521

1995
[35]

Krause, Highest weight categories and recollements, Ann

H. Krause, Highest weight categories and recollements, Ann. Inst. Fourier (Grenoble)67(2017), no. 6, 2679–2701; MR3742477

2017
[36]

Krause,Homological theory of representations, Cambridge Studies in Advanced Mathemat- ics, 195, Cambridge Univ

H. Krause,Homological theory of representations, Cambridge Studies in Advanced Mathemat- ics, 195, Cambridge Univ. Press, Cambridge, 2022; MR4327095

2022
[37]

Lascoux, B

A. Lascoux, B. Leclerc and J.-Y. Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys.181(1996), no. 1, 205–263; MR1410572

1996
[38]

Losev, Quantum category O vs affine Hecke category, arXiv preprint: 2310.03153 (2023)

I. Losev, Quantum category O vs affine Hecke category, arXiv preprint: 2310.03153 (2023)

2023 arXiv
[39]

P. P. Martin and D. J. Woodcock, The partition algebras and a new deformation of the Schur algebras, J. Algebra203(1998), no. 1, 91–124; MR1620713

1998
[40]

Williamson, Parity sheaves and the Hecke category, inProceedings of the International Congress of Mathematicians—Rio de Janeiro 2018

G. Williamson, Parity sheaves and the Hecke category, inProceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. I. Plenary lectures, 979–1015, World Sci. Publ., Hackensack, NJ, ; MR3966750 40 C. Hone: School of Mathematics and Statistics, Universi...

2018

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [2]

A remarkable functor on $G$-modules

J. Baine, T. Fell, A. Romanov, A. Sherman, and G. Williamson, A remarkable functor on G-modules. ArXiv preprint 2505.07144 (2025)

work page Pith review arXiv 2025

[2] [10]

Towards higher Frobenius functors for symmetric tensor categories

Coulembier, Kevin, and Johannes Flake. ”Towards higher Frobenius functors for symmetric tensor categories.” arXiv preprint arXiv:2405.19506 (2024)

work page Pith review arXiv 2024

[3] [1]

J. L. Alperin,Local representation theory, Cambridge Studies in Advanced Mathematics, 11, Cambridge Univ. Press, Cambridge, 1986; MR0860771

work page 1986

[4] [3]

Balmer, The spectrum of prime ideals in tensor triangulated categories, J

P. Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math.588(2005), 149–168; MR2196732

work page 2005

[5] [4]

J. W. Barrett and B. W. Westbury, Spherical categories, Adv. Math.143(1999), no. 2, 357– 375; MR1686423

work page 1999

[6] [5]

D. J. Benson,Representations of elementary abelianp-groups and vector bundles, Cambridge Tracts in Mathematics, 208, Cambridge Univ. Press, Cambridge, 2017; MR3585474

work page 2017

[7] [6]

Brundan, On the definition of Heisenberg category, Algebr

J. Brundan, On the definition of Heisenberg category, Algebr. Comb.1(2018), no. 4, 523–544; MR3875075

work page 2018

[8] [7]

Brundan, A

J. Brundan, A. Savage and B. Webster, Heisenberg and Kac-Moody categorification, Selecta Math. (N.S.)26(2020), no. 5, Paper No. 74, 62 pp.; MR4169511

work page 2020