Representation ring of Levi subgroups versus cohomology ring of flag varieties II

Sean Rogers; Shrawan Kumar

arxiv: 1907.10089 · v1 · pith:5AWJ2USCnew · submitted 2019-07-19 · 🧮 math.RT · math.AT· math.GR

Representation ring of Levi subgroups versus cohomology ring of flag varieties II

Shrawan Kumar , Sean Rogers This is my paper

Pith reviewed 2026-05-24 18:43 UTC · model grok-4.3

classification 🧮 math.RT math.ATmath.GR

keywords representation ringLevi subgroupcohomology ringflag varietiessymplectic groupsisotropic Grassmanniansring homomorphismsinjectivity

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The pith

A homomorphism from the representation ring of Sp(2k) to the cohomology of the isotropic Grassmannian becomes injective as n tends to infinity.

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

This paper studies a ring homomorphism ξ_{n,k} that sends a subring of the complexified representation ring of the symplectic group Sp(2k) to the cohomology ring with complex coefficients of the isotropic Grassmannian IG(n-k, 2n) inside the larger symplectic group Sp(2n). The authors prove that this map is injective once the ambient dimension parameter n is taken to infinity while keeping k fixed. The construction originates from a general setup for reductive groups and their parabolic subgroups, here specialized to the defining representation. Similar injectivity results are established for the odd orthogonal groups.

Core claim

The paper asserts that the C-algebra homomorphism ξ_{n,k} from Rep^C_{ω1-poly}(Sp(2k)) to H^*(IG(n-k, 2n), C) is injective when n tends to infinity keeping k fixed. Analogous statements hold for the odd orthogonal groups.

What carries the argument

The ring homomorphism ξ_{n,k} introduced via the general construction ξ^P_λ using the defining representation V(ω1) of Sp(2n).

If this is right

The kernel of ξ_{n,k} is trivial for sufficiently large n.
The image of the representation subring embeds into the cohomology ring.
This holds similarly for odd orthogonal groups and their corresponding flag varieties.

Where Pith is reading between the lines

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

This injectivity may imply that the stable cohomology is generated by classes coming from representations of the smaller group.
One could test the result by explicit computation of both rings for small fixed k and increasing n.
The result suggests a stabilization phenomenon linking algebraic and geometric invariants.

Load-bearing premise

The ring homomorphism ξ^P_λ is well-defined when applied to the subring Rep^C_λ-poly(L) with V(λ) the defining representation.

What would settle it

Compute the kernel of ξ_{n,k} explicitly for a sequence of increasing n with fixed k and check if it remains trivial; a non-zero element that maps to zero cohomology class would disprove the claim.

read the original abstract

For any reductive group G and a parabolic subgroup P with its Levi subgroup L, the first author in [Ku2] introduced a ring homomorphism $ \xi^P_\lambda: Rep^\mathbb{C}_{\lambda-poly}(L) \to H^*(G/P, \mathbb{C})$, where $ Rep^\mathbb{C}_{\lambda-poly}(L)$ is a certain subring of the complexified representation ring of L (depending upon the choice of an irreducible representation $V(\lambda)$ of G with highest weight $\lambda$). In this paper we study this homomorphism for G=Sp(2n) and its maximal parabolic subgroups $P_{n-k}$ for any $1\leq k\leq n$ (with the choice of $V(\lambda) $ to be the defining representation $V(\omega_1) $ in $\mathbb{C}^{2n}$). Thus, we obtain a $\mathbb{C}$-algebra homomorphism $ \xi_{n,k}: Rep^\mathbb{C}_{\omega_1-poly}(Sp(2k)) \to H^*(IG(n-k, 2n), \mathbb{C})$. Our main result asserts that $ \xi_{n,k}$ is injective when n tends to $\infty$ keeping k fixed. Similar results are obtained for the odd orthogonal groups.

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 injectivity for the ξ homomorphism on representation rings of fixed-rank Levis inside Sp(2n) and odd orthogonal groups as n goes to infinity.

read the letter

The main new statements are the injectivity of ξ_{n,k} : Rep^C_{ω1-poly}(Sp(2k)) → H^*(IG(n-k,2n),C) in the limit n→∞ with k fixed, plus the parallel claim for the odd orthogonal case. These are not in the earlier Kumar paper that defined the map, so the contribution is the verification for these classical groups and the asymptotic regime. The construction itself is taken from the prior work and applied to the defining representation, which keeps the argument focused. The abstract is clear about the target and the limit statement. The stress-test note finds no internal contradiction or hidden circularity in the setup, which matches what is visible here. The main limitation is that the provided text is only the abstract; without the actual proof steps or any explicit verification for small k, it is hard to judge how much new technique is required versus direct application of the earlier homomorphism. The result sits inside a narrow subfield of representation theory and cohomology of homogeneous spaces. A reader already working on maps from representation rings to cohomology rings of flag varieties will find the statements useful for reference. It is not broad enough to change how people outside that circle think about either ring. I would send it to a referee who knows the prior Kumar paper and the classical-group calculations, because the claim is concrete and the setup is reproducible from the cited definition. If the proofs check out, it is a solid incremental result worth recording.

Referee Report

0 major / 3 minor

Summary. The paper studies the ring homomorphism ξ^P_λ : Rep^C_{λ-poly}(L) → H^*(G/P, C) from prior work [Ku2], specialized to G = Sp(2n) with maximal parabolic P_{n-k} and V(λ) the defining representation. It defines ξ_{n,k} : Rep^C_{ω1-poly}(Sp(2k)) → H^*(IG(n-k, 2n), C) and asserts that ξ_{n,k} is injective in the limit n → ∞ with k fixed; analogous statements are proved for odd orthogonal groups.

Significance. If the injectivity holds, the result gives a representation-theoretic description of the cohomology rings of these flag varieties in the stable range, connecting fixed-rank representation rings to an asymptotic geometric invariant. The construction reuses the prior homomorphism without new definitions and obtains parallel statements for two classical series.

minor comments (3)

Abstract: the statement of the main result is clear, but the abstract supplies no indication of the proof strategy or the key estimates used to pass to the n → ∞ limit.
The manuscript relies on the well-definedness of ξ^P_λ on the indicated subring from [Ku2]; a short self-contained paragraph recalling the properties actually invoked would improve readability.
Notation: the transition from the general ξ^P_λ to the specialized ξ_{n,k} is introduced without an explicit equation relating the two; adding a displayed equation would clarify the specialization.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. The referee's description of the main result (injectivity of ξ_{n,k} in the stable range n → ∞ with k fixed, and the analogous statement for odd orthogonal groups) is accurate. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

Minor self-citation for map construction; central injectivity independent

full rationale

The homomorphism ξ^P_λ is taken from prior work [Ku2] by the first author and applied to the defining representation of Sp(2n) and its Levi subgroups. The main result is the new statement that the induced map ξ_{n,k} becomes injective in the limit n→∞ with k fixed (and analogous statements for odd orthogonal groups). No equation in the provided text reduces the injectivity claim to a fitted parameter, a self-definition, or a quantity defined in terms of the result itself. The self-citation supports the well-definedness of the input map but is not load-bearing for the asymptotic injectivity proof, which has independent content. This matches the default expectation of score 0-2 for papers whose central claim does not collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the prior construction of the homomorphism ξ^P_λ and the definition of the λ-poly subring; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption The ring homomorphism ξ^P_λ : Rep^C_λ-poly(L) → H^*(G/P,C) is well-defined as introduced in [Ku2]
The current paper invokes this map for the specific choice of G=Sp(2n), P=P_{n-k}, and V(λ)=V(ω1) without re-deriving its properties.

pith-pipeline@v0.9.0 · 5759 in / 1370 out tokens · 24575 ms · 2026-05-24T18:43:35.220806+00:00 · methodology

Representation ring of Levi subgroups versus cohomology ring of flag varieties II

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)