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Eigenweights for arithmetic Hirzebruch Proportionality

Tony Feng

arXiv:2601.23245v2math.RTmath.NT

TL;DR

The paper addresses the missing general calculation of eigenweights governing Arithmetic Hirzebruch Proportionality. It uses algebraic combinatorics and symmetric-group representation theory to determine eigenweights for all classical groups.

  • Problem

    Arithmetic Hirzebruch Proportionality had eigenweights computed only in examples, leaving their general determination open.

  • Method

    The paper connects eigenweights to symmetric-group representation theory and derives uniform formulas for the relevant classical-group cases.

  • Results

    The paper determines eigenweights for all classical groups, including previously uncalculated cases for types A, B, C, and D.

  • Takeaways & Limitations

    The eigenweights admit a clean, uniform description in terms of symmetric-group representation theory.

  • Takeaways & Limitations

    The paper restricts attention to a specified Casimir-derived format of η and does not discuss exceptional groups.

Abstract

from arXiv · show

Prior work of Feng--Yun--Zhang established a (Higher) Arithmetic Hirzebruch Proportionality Principle, expressing the arithmetic volumes of moduli stacks of shtukas in terms of differential operators applied to $L$-functions. This formula involves certain "eigenweights" which were calculated in simple cases by Feng--Yun--Zhang, but not in general. We document work of a (custom) AI Agent built upon Gemini Deep Think, which employs tools from algebraic combinatorics to connect these eigenweights to the representation theory of symmetric groups, and then determines them for all classical groups.

1. Introduction

The paper studies eigenweights in Arithmetic Hirzebruch Proportionality, where arithmetic volumes are expressed through differential operators applied to L-functions. It gives a uniform symmetric-group representation-theoretic description for classical groups, including previously unresolved cases, while identifying a commutativity failure in type D.

  • Arithmetic Hirzebruch Proportionality: Arithmetic Hirzebruch Proportionality relates arithmetic volumes of Chern classes on arithmetic moduli spaces to differential operators applied to L-functions.The principle applies in both number-field and function-field settings.
  • Definition of eigenweights: Eigenweights are eigenvalues of a local operator ∇η,µ acting on the first augmentation-graded piece V of the Weyl-invariant cohomology ring.The paper restricts to classes η constructed from a Casimir element so that the operator preserves grading.
  • Scope and limitation: The paper excludes exceptional groups from closed-form discussion and shows that eigenweight operators need not commute for different minuscule coweights in type D2m.In particular, the relevant matrix is not diagonal in general.
  • General linear groups: Theorem 1.3.2 determines the previously unavailable closed-form eigenweights for GLn and minuscule coweights (1m, 0n−m) with m≥3 using symmetric-group representation theory.Earlier work had not calculated these cases in closed form, although the resulting formulas are more complex than prior examples.
  • Symplectic and orthogonal groups: Theorems 1.3.6 and 1.3.8 determine eigenweights for the spin minuscule coweight of PSp2n and the spinor minuscule coweights of PSO2n.For PSp2n the formulas use the partition ρn=(n,n−1,…,1), while for PSO2n they depend on the parity of n and involve δn=(n−1,…,1,0).

2. Declaration of AI Usage

The paper’s core mathematics was generated by Aletheia, an internal reasoning agent built on Gemini Deep Think, and rewritten by the human author. A Human–AI Interaction Card documents the process, raw outputs, and fully correct queries for Types A, C, and D.

  • Authorship and AI Usage: Aletheia generated the paper’s core mathematical content, while the human author rewrote the paper from its output.Aletheia is described as an internal reasoning agent built upon Gemini Deep Think and codenamed at Google DeepMind.
  • Authorship and AI Usage: The Human–AI Interaction Card summarizes the interaction and links to raw model outputs for comparison with the paper.The comparison is intended to let readers verify the relationship between the paper and the original outputs.
  • Mathematical Content: The expression i=1 (2i−1)n−i in (1.3.7) is identified as the dimension of the symmetric-group irreducible representation associated with the partition δn.This explanation connects the formula’s factor to representation-theoretic data.
  • Interaction Record: The documented workflow queried eigenweights for Types A, C, and D, with each recorded Aletheia response marked fully correct.The card records separate human prompts and correctness assessments for all three types.

3. Eigenweights for type A

For type A, the section specializes to G = GL_n and identifies the relevant Weyl-group action with the symmetric-group action on a polynomial ring. Its invariant ring is the ring of symmetric polynomials in the variables x_1, …, x_n.

  • The setup fixes n ≥ 2 and takes G = GL_n, with R identified as Q[x_1, …, x_n] and deg x_i = 2.
  • The Weyl group W is identified with the symmetric group S_n, acting on R by permuting the variables x_i.
  • The invariant ring R^W = H^*(BG) is the ring of symmetric polynomials in X = {x_1, …, x_n}.

3.1. Setup.

The setup fixes a minuscule coweight for GL_n and identifies the associated invariant-theoretic and geometric data. It also introduces power-sum elements, including t_μ and η, used in the subsequent construction.

  • Power-sum notation: The power sums p_i form a basis for V = I/I^2, where I is the augmentation ideal.
  • Invariant-theoretic setup: For 1 ≤ m < n, the minuscule coweight is μ = (1^m, 0^{n−m}), and H*(BP_μ) ≅ H*(BL_μ) ≅ R^{W_μ} consists of polynomials separately symmetric in variable sets A and B.Here A = {x_1, …, x_m} and B = {x_{m+1}, …, x_n}.
  • Invariant-theoretic setup: The associated Levi subgroup is L_μ ≅ GL_m × GL_{n−m}, with Weyl group W_μ = S_m × S_{n−m}.
  • Geometric setup: The flag variety G/P_μ is the Grassmannian Gr(m, n), of dimension D_μ = m(n − m), and N is defined as D_μ + 1.
  • Power-sum notation: The setup specifies t_μ = p_1(A) and η = p_1(A)^N.

3.2. Proof of Theorem 1.3.2.

The proof identifies power sums as eigenvectors and derives their eigenweights through Schur-polynomial identities and symmetric-group character theory. A rectangular-partition lemma and the Murnaghan–Nakayama rule reduce the character calculation to hook partitions, completing Theorem 1.3.2.

  • Eigenvectors and eigenweights: Power sums p1(X), …, pn(X) form homogeneous basis elements of V = I/I2 and therefore are eigenvectors of ηµ.The eigenweights εk(Ω, µ) are defined by ηµ(pk(X)) ≡ εk(Ω, µ)pk(X) (mod I2).
  • Schur-polynomial reduction: The rectangular-partition lemma evaluates the relevant Schur-polynomial expression as sλ−Π(X) when λ contains Π, and as zero otherwise.Here Π is the m × (n − m) rectangular partition.
  • Symmetric-group characters: Frobenius character formulas convert the eigenweight calculation into symmetric-group character values χπ((k)) for the cycle type νk = (k−1, 1^N).The resulting expression gives a formula for the eigenweight associated to pk.
  • Murnaghan–Nakayama reduction: χπ((k)) vanishes unless π is a hook partition πj(k) = (k − j, 1^j), in which case χπj((k)) = (−1)^j.This follows from the Murnaghan–Nakayama rule because nonzero character values require π itself to be a border strip.
  • Completion of the proof: The hook-character reduction, together with the constraint len(π) = j + 1 ≤ m, yields the desired eigenweight formula and completes the proof of Theorem 1.3.2.The proof concludes immediately after applying Lemma 3.2.8 to the preceding eigenweight expression.

3.3. Examples.

The examples apply Theorem 1.3.2 to compute eigenweights for m = 1 and m = 2 using the Murnaghan–Nakayama rule. These calculations recover the previously known results in FYZ25a, Propositions 4.4.1 and 5.4.1.

  • m = 1: For n ≥ 2 and m = 1, only j = 0 contributes, with π_j = (k) and π_j(k) + (n − m)m = (n + k − 1).The resulting character evaluation is (−1)^(n−1), recovering FYZ25a, Proposition 4.4.1.
  • m = 2: For n ≥ 3 and m = 2, assuming k ≥ 2, the relevant data are Π = (n − 2, n − 2) and ν_k = (k − 1, 1^{2(n−2)+1}).The case k = 1 is similar but simpler, and (−1)^(N−1) = 1.
  • m = 2: For m = 2, only j = 0 and j = 1 contribute, and the resulting character terms are evaluated with the Murnaghan–Nakayama rule.The contributing border strips are horizontal, with the first term allowing a second-row strip when k < n and the second term using the second row.
  • m = 2: The relevant identity character values are χ(k+n−2,n−k−1)(Id) = (2n − 3)!(2k) / ((k + n − 1)!(n − k − 1)!) and χ(k+n−3,n−k)(Id) = (2n − 3)!(2k − 2) / ((k + n − 2)!(n − k)!).Substitution yields equation (3.3.3), whose simplification agrees with (1.3.2), recovering FYZ25a, Proposition 5.4.1.

4. Eigenweights for type C

For type C_n, the eigenweight problem is reduced to symmetric polynomials for the Levi subgroup and analyzed through the Gysin map. The calculation uses parity constraints on Schur polynomials and symmetric-group character values to complete the eigenweight determination.

  • Type C setup: For G = PSp2n, the Weyl group is the signed-permutation group of type C_n, while the Levi Weyl group is S_n acting by permutations.The invariant ring for G is generated by power sums in x_1^2, …, x_n^2, whereas the Levi invariant ring consists of symmetric polynomials in x_1, …, x_n.
  • Eigenvector reduction: The power sums in x_1^2, …, x_n^2 form a homogeneous degree basis for V = I/I^2 and therefore are eigenvectors for the relevant differential operator.The associated eigenweights are defined by the operator’s action on these basis elements.
  • Gysin-map calculation: The Gysin map sends a Schur polynomial s_λ(X) to zero unless λ + δ_n is purely odd; in the nonzero case, λ has the form 2π + ρ_n.The proof uses projection onto the sign-change invariants, which extracts monomials with purely even exponents.
  • Symmetric-group reduction: The resulting power-sum calculation is concentrated on partitions λ = 2π + ρ_n with |π| = k, reducing the eigenweight computation to symmetric-group character values.Only hook-shaped partitions π_j(k) = (k−j, 1^j) contribute, with character value χ_πj(k)((k)) = (−1)^j.
  • Conclusion: The character computation completes the type C eigenweight determination stated in Theorem 1.3.6.The conclusion follows after projecting the expression to V = I/I^2 and substituting the contributing hook partitions.

5. Eigenweights for type D

For type D, the section considers G = PSO_{2n} with n ≥ 2 and the polynomial ring R = Q[x_1, …, x_n], where each variable has degree 2.

  • Type D setup: The type D setting is G = PSO_{2n} with n ≥ 2.
  • Type D setup: The associated ring is R = Q[x_1, …, x_n].
  • Type D setup: Each generator x_i has degree 2.

5.1. Setup.

For type D_n, the setup uses the even signed-permutation Weyl group and its invariant ring, then focuses on computing eigenweights for a spinor coweight. The relevant Levi subgroup has type A_{n−1}, with Weyl group S_n and symmetric-polynomial invariants.

  • Weyl-group invariants: The Weyl group W(D_n) consists of even signed permutations, and its invariant ring is generated by the indicated polynomial invariants together with the Pfaffian.The supplied setup identifies the Pfaffian as having degree 2n.
  • Minuscule coweights: The minuscule coweights comprise the vector coweight ω∨_1 and two spinor coweights, ω∨_n and its outer-automorphism partner.The two spinor representations are exchanged by the order-2 Dynkin-diagram automorphism.
  • Target eigenweights: The computation focuses on the spinor coweight μ := ω∨_n because the two spinor coweights are exchanged by an automorphism and therefore have equal eigenweights.The vector coweight eigenweights were computed previously, while the spinor case is described as substantially more difficult.
  • Levi reduction: For μ, the Levi subgroup L_μ has type A_{n−1}, its Weyl group is S_n, and its invariant ring consists of symmetric polynomials in X.The setup also defines D_μ as the dimension of G/P_μ and N := D_μ + 1.
  • Invariant-ring operator: The relevant operator maps the D_n invariant ring R_W to the Levi invariant ring R_{W_μ}, with the displayed linear power-sum identity providing part of the setup.The supplied passage records the identity 2(x_1 + ... + x_n) = 1/2 p_1(X).

5.2. Proof of Theorem 1.3.8.

The proof determines eigenweights by reducing the relevant integration calculations to symmetric-group character and Schur-polynomial combinatorics. It identifies a basis of eigenvectors, with an exceptional even-dimensional degree where the action is represented by a matrix whose off-diagonal entries can be nonzero.

  • Eigenvector decomposition: The quotient V = I/I^2 is one-dimensional in each nonzero degree except degree 2n when n = 2m, where V_2n is spanned by p^(2)_m and Pf.For odd n, the classes p^(2)_1, ..., p^(2)_{n−1}, Pf form a homogeneous basis and hence are eigenvectors.
  • Exceptional even case: B_m and C_m can be non-zero, so the exceptional even-dimensional action need not be diagonal in the basis p^(2)_m, Pf.The proof explicitly computes the matrix entries and addresses the question raised in.
  • Combinatorial calculation: The integration calculation uses the even-sign-change group K, the bialternant formula, Frobenius character formula, and Lemma 5.2.2 to isolate the contributing parity cases.A character term contributes only when all γ_j have the same parity, while the odd case uses Pf = x_1x_2 · · · x_n.
  • Combinatorial calculation: The odd Case O contributes modulo I^2 only when n = 2m, k = m, π = ∅, and λ = δ_n + (1^n), producing the Pf component.For |π| > 0, the resulting scalar multiple of Pf s_π(X^(2)) vanishes modulo I^2 because Pf ∈ I.
  • Eigenweights: For odd n, the proof obtains the eigenweight formulas for all basis elements, while for n = 2m the same formulas hold for ε_k when k ≠ m.The exceptional degree V_2n is instead governed by the displayed matrix on the basis involving p^(2)_m and Pf.

5.3. Examples.

The examples compute eigenweight-related quantities using character values, skew-shape identities, and the Hook Length formula. In the concluding example, the eigenweights are 4 and 2, agreeing with prior results by triality.

  • The example uses χ(4,3,2,1)((3, 17)) = −48 in its computation.
  • The upper-right entry is evaluated by rewriting skew shapes as partitions and applying the Hook Length formula.The shapes become (6, 1) and (4, 3) after subtracting (1, 1, 1).
  • −1/8 is obtained for B2 from the difference 2−6 · (6 −14).
  • The eigenweights are 4 and 2, matching results (1.3.5) as required by triality of the D4 root system.
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