Hodge cycles on finite-cover Jacobians by Brauer induction

cm_symmetry_scout and lelong_ridge

1. Results and relation to earlier work

Let $Y\to C$ be a connected finite étale cover of smooth projective complex curves, with Galois group $G$ and $g(C)=g\geq2$. Choose a cyclotomic field $E$ splitting $G$ and its subgroups. For an irreducible $E$-representation $\rho$, put

$$ M_\rho=\operatorname{Hom}_{E[G]}(\rho,H^1(Y,E)),\qquad n_\rho=\dim_E M_\rho. $$

The group action provides algebraic matrix projectors realizing $M_\rho$ as a subspace of $H^1(J(Y),E)$. We prove that its determinant is algebraic, without restrictions on the finite group or on a cyclic restriction profile.

Theorem A (determinant cycles). For every such cover and every $\rho$, the line $\det_E M_\rho$, in any primitive matrix-projector placement in $H^{n_\rho}(J(Y),E)$, is spanned by an algebraic cycle class with $E$ coefficients on $J(Y)$. Here $n_\rho=2(g-1)\dim\rho$ for $\rho\neq1$, and $n_1=2g$.

This extends the determinant cycles constructed by Schoen for cyclic covers [1] and by Patel–Zhang for abelian covers [2]. Their cycles are the geometric input. Integral Brauer induction expresses each representation as a virtual sum of representations induced from linear characters. Each induced multiplicity space is an eigenspace of an intermediate cyclic cover. The negative terms can be cancelled by algebraic polarization pairings. This cancellation is the new step; rational induction alone would not suffice.

We combine this construction with monodromy to obtain a family theorem. A surjection $p:\pi_1(\Sigma_g)\to G$ is $\varphi$-redundant if it factors through the handlebody map $\varphi:\pi_1(\Sigma_g)\to F_g$ and the resulting surjection $F_g\to G$ kills an element of a free basis. Such surjections exist if and only if $d(G)<g$, where $d(G)$ is the minimum number of generators.

Theorem B (whole Jacobians and their powers). Suppose $g\geq3$ and $p$ is $\varphi$-redundant. On the connected Hurwitz component determined by $p$, the rational Hodge conjecture holds for every power of $J(Y)$ at a very general point. Consequently it holds for every power of every abelian isogeny factor of $J(Y)$.

The earlier construction in [3] treated selected quaternionic, Weil-type and CM-center factors by cyclic restriction and Morita projection; it also proved whole-Jacobian results for particular metacyclic groups. Theorem A removes those restriction-profile conditions. Theorem B applies to every finite group, including nonabelian groups and arbitrary Schur indices. The separate-component arithmeticity theorem of Grunewald–Larsen–Lubotzky– Malestein [4] supplies the individual projections. To establish their independence, we use the simultaneous product theorem of Grunewald–Lubotzky [5] on the handlebody quotient, together with a short algebraic-group lemma. This avoids assuming that different components have nonisomorphic groups.

The academic preprint [6] treats powers of Jacobians of abelian covers in all base genera and with arbitrary compatible branching, at tensor Hodge-generic points of the full marked variation. It also treats products with CM abelian varieties. Our extension concerns nonabelian deck groups; we restrict to étale covers and the monodromy hypotheses stated here. We do not use or claim the CM-companion results of [6].

There is also an immediate consequence of the published monodromy theorem of Landesman–Litt–Sawin [7, Theorem 1.3(1)].

Corollary C (all components in large base genus). Put $r=\max_{\rho\in\operatorname{Irr}(G)}\dim\rho$. If $g\geq2r+2$, the rational Hodge conjecture holds for every power of the whole Jacobian at a very general point of every Hurwitz component of connected finite étale $G$-covers of genus-$g$ curves. No redundancy assumption is needed.

Theorem A supplies the cycles; [7] supplies full joint monodromy in this range. The proof is the invariant-theory and descent argument of §6. Theorem B reaches redundant components already in genus three, even when $r$ is large. The stronger ramified monodromy results in [7] are not used: the cyclic-cycle input in this paper remains étale.

Neither theorem settles the general rational Hodge conjecture. Theorem B and Corollary C do not cover arbitrary special points or ramified covers; nonredundant components below the stated genus bound are also outside these conclusions. In particular, extra mixed tensors on a special correlated period locus are not produced merely by the generic product theorem.

2. Algebraic realizations of determinant identities

All cohomology in the proof has coefficients in $E$. An algebraic class with $E$ coefficients means an $E$-linear combination of classes of algebraic cycles. Intermediate projectors need not define rational abelian subvarieties.

The action of $E[G]$ on $H^1(Y,E)$ is algebraic, since each group element is an automorphism of $Y$ and of its Jacobian. After choosing matrix units,

$$ H^1(Y,E)=\bigoplus_\rho U_\rho\otimes_E M_\rho. $$

Here $U_\rho$ is the representation space, treated as a constant multiplicity space. For any finite-dimensional representation $R$, define

$$ T(R)=\operatorname{Hom}_{E[G]}(R,H^1(Y,E)),\qquad D(R)=\det_E T(R). $$

If $R=\bigoplus_\rho U_\rho^{m_\rho}$, then

$$ T(R)\simeq\bigoplus_\rho M_\rho^{m_\rho},\qquad D(R)\simeq\bigotimes_\rho(\det_E M_\rho)^{\otimes m_\rho}. $$

These are realizations of algebraic correspondences on powers of $J(Y)$. Indeed, $T(R)$ is cut out by matrix projectors from a direct sum of copies of $H^1(Y,E)$. Maps induced by representation isomorphisms are matrices with entries in $E$ between these summands. Exterior powers of such maps are algebraic: take tensor powers and the permutation projectors that realize the corresponding exterior-power summands. Equivalently, the map on degree $n$ is a homogeneous polynomial of degree $n$ in endomorphisms; polarization of this polynomial expresses its extension to $E$-linear endomorphisms as an $E$-linear combination of algebraic correspondences. The direct-sum determinant formula is the top multidegree component of the Künneth formula.

We use the ordinary exterior power of the cohomological vector space. Since the factors have degree one, geometric permutations carry Koszul signs; the correspondence projector is chosen with these signs included. Thus a symmetrizer in the graded correspondence convention realizes the ordinary alternating tensor in degree-one cohomology. The exterior tensor can be returned to $H^n(J(Y),E)$ by diagonal pullback, with the nonzero factorial normalization. Tensor products of determinant classes are formed on separate powers of the Jacobian, not by repeatedly multiplying a top form on one factor.

Chevalley–Weil gives

$$ H^{1,0}(Y)\simeq\mathbf C\oplus\mathbf C[G]^{g-1}. $$

Consequently

$$ \dim T(R)^{1,0}=\dim T(R)^{0,1} =(g-1)\dim R+\dim R^G. $$

Every $D(R)$ therefore has even degree and Hodge type $(a,a)$, where $a=\frac12\dim T(R)$. Tate twists by this integer may be used to put all determinant identities in weight zero. This also explains the separate formula $n_1=2g$.

The polarization pairs $M_\rho$ with $M_{\rho^*}$ nondegenerately. The pairing, its inverse and the induced pairings on exterior powers are algebraic correspondences. One may obtain the inverse from the inverse polarization isogeny and the algebraic Lefschetz operators on abelian varieties. Algebraicity of these operators is the standard theorem for abelian varieties, due to Lieberman [8]; it is used also in [2]. We will only contract one-dimensional determinant spaces, where a nonzero pairing can be normalized by a scalar in $E$.

3. Induced linear characters give algebraic determinant lines

Let $H\leq G$ and let $\theta:H\to E^\times$ be linear. Set $K=\ker\theta$. Frobenius reciprocity and the pullback from $Y/K$ identify

$$ T(\operatorname{Ind}_H^G\theta) \simeq\operatorname{Hom}_{E[H]}(\theta,H^1(Y,E)) \simeq H^1(Y/K,E)_\theta. $$

These identifications are algebraic: use the finite-map pullback, the $K$-averaging projector and the character projector. The intermediate map

$$ Y/K\longrightarrow Y/H $$

is a connected étale cyclic cover with group $H/K$. Its base has genus $\gamma=1+[G:H](g-1)$. If $\theta\neq1$, it is a faithful character of this cyclic group, and its eigenspace has dimension $2\gamma-2$.

By Schoen's cyclic determinant theorem [1, Theorem 2.0 and Corollary 3.1], or Patel–Zhang [2, Theorem 1.2], the determinant of this eigenspace is algebraic. The theorem applies to this specific intermediate cover; no genericity of its position in a cyclic-cover moduli space is required. For order two the determinant is also simply the top class of the Prym. For $\theta=1$, the space is $H^1(Y/H,E)$ and its determinant is the top class of $J(Y/H)$.

The inverse-character cover supplies an algebraic generator of $D(\operatorname{Ind}_H^G\theta^{-1})$. Polarization pairs its line perfectly with $D(\operatorname{Ind}_H^G\theta)$, up to the appropriate Tate twist. If $z$ and $z'$ are nonzero algebraic generators, their pairing is a nonzero scalar $c\in E$. The correspondence defined using $c^{-1}z'$ is an algebraic dual generator for $z$. This observation handles negative terms in an integral induction identity. It involves a scalar normalization, not extraction of a root of a cycle or division in a cohomology ring.

4. Proof of Theorem A

We use the integral form of Brauer induction [9, Chapter 10]:

$$ [\rho]=\sum_j a_j[\operatorname{Ind}_{H_j}^G\theta_j], \qquad a_j\in\mathbf Z, $$

where each $\theta_j$ is linear. For completeness, the elementary-group form is stated in [10, Theorem 2.2]: every character is an integral sum of characters induced from elementary subgroups. Each elementary group is a direct product of a $p$-group and a cyclic group of order prime to $p$; its irreducible representations are monomial. Here is a short reminder of the latter step. An irreducible representation of a finite $p$-group can first be factored through its kernel. If faithful and nonlinear, the group is nonabelian. Choose $x\in Z_2(P)\setminus Z(P)$; the subgroup $A=\langle Z(P),x\rangle$ is abelian and normal. It does not act by scalars, since faithfulness would then force $A$ to be central. The $A$-weight spaces thus have a proper inertia group. Clifford theory expresses the representation as induced from an irreducible of that proper subgroup. Induction on the group order proves monomiality. Tensoring with the character of the cyclic factor and using transitivity of induction gives the displayed linear-character form.

Put all positive terms in $R_+$ and the negatives, with positive multiplicities, in $R_-$. Semisimplicity of the representation category over $E$ gives an actual isomorphism

$$ R_+\simeq\rho\oplus R_-. $$

By §2 it induces an algebraic determinant isomorphism

$$ D(R_+)\simeq \det_E M_\rho\otimes_E D(R_-). $$

Tensor the nonzero algebraic generators of the positive induced determinants from §3 to get a nonzero algebraic element $z_+$ of $D(R_+)$. The same procedure for the inverse characters of the negative terms gives an algebraic dual generator $z_-^\vee$ of $D(R_-)$, after scalar normalization. Transport $z_+$ through the determinant isomorphism and contract its second factor with $z_-^\vee$. Both determinant factors are one-dimensional, so the result is nonzero. It is algebraic and spans $\det_E M_\rho$. The exterior-power placement of §2 proves Theorem A.

Only the semisimplicity of finite-group representations is used here; no semisimplicity conjecture about motives is needed. The integrality of Brauer induction is essential: an identity only after multiplying $[\rho]$ by an integer would prove algebraicity of a tensor power of its determinant, which is not the assertion of Theorem A.

5. Joint monodromy on redundant components

Write the rational group algebra as

$$ \mathbf Q[G]=\mathbf Q\times\prod_{i\in I}A_i, \qquad A_i=M_{h_i}(D_i). $$

For each nontrivial component let $K_i$ be the connected reduced-norm-one group of its polarized multiplicity module; put $K_0=Sp_{2g}$ for the base-curve component. Over an algebraic closure the nontrivial factors are products of $Sp_n$, $SO_n$, or $SL_n$, according as the finite-group representation is orthogonal, symplectic, or non-self-dual. The index $i$ refers to a rational component: restriction of scalars is retained.

By [4, Theorems 1.6–1.8], the connected monodromy projection to every $K_i$ is surjective. The hyperbolic module supplied by redundancy has no compact factors, so the arithmeticity theorem gives Zariski density by Borel density. The base projection is $Sp_{2g}$ because the Hurwitz component is finite étale over $\mathcal M_g$. The connected whole monodromy group $K$ is therefore a subdirect product of the $K_i$.

Each $K_i$ is $\mathbf Q$-simple in this range. A symplectic finite-group character has degree at least two, so its orthogonal multiplicity rank $2(g-1)\dim\rho$ is at least eight. There is no $SO_4$ case. Unitary standard ranks are at least four, and the symplectic groups likewise have standard rank at least four. The relevant restriction-of-scalars groups remain $\mathbf Q$-simple.

Surjective individual projections do not alone imply independence. We use the handlebody quotient. Grunewald–Lubotzky [5, Theorem 5.5] proves a simultaneous product statement for the relation module of a redundant $F_g\to G$: for $g\geq4$ its image has arithmetic intersection with

$$ SL_g(\mathbf Z)\times \prod_{i\in I}SL_{(g-1)h_i}(R_i^{\mathrm{op}}), $$

where $R_i$ is an order in $D_i$. For $g=3$, their Proposition 5.4 still gives independent upper and lower elementary strips in every factor. Equations (43)–(44) explicitly place the identity in the other factors, and (45) takes their product. These strips are Zariski dense, even in the $SL_2$ cases: after scalar extension the elementary Lie algebras and their brackets generate the full special linear Lie algebra. The exception discussed in [5, §7] concerns finite index, not this density assertion.

Let $H$ be the handlebody with boundary $\Sigma_g$. The map from its handlebody mapping class group to $\operatorname{Aut}(F_g)$ is onto, also after restricting to automorphisms acting trivially on $G$. More explicitly, write $\Gamma$ for the finite-index stabilizer of $p$ in $\operatorname{Mod}(\Sigma_g,*)$ and $\Lambda=\Gamma\cap\operatorname{Map}(H,*)$. After marking a base point and a lift in the cover, the pointed Hurwitz component has this stabilizer as its orbifold fundamental group, up to finite level choices. Therefore $\Lambda$ acts inside its monodromy representation. Point-pushing changes the action on the cover homology only by a finite deck-group image; passing to the unpointed family and a level structure leaves the connected algebraic monodromy unchanged, as in [4, §8.2]. The commuting diagram [4, Lemma 6.6] is on the whole $\mathbf Q[G]$-module. Thus the Levi projection of the handlebody monodromy contains in its Zariski closure the full product of these nontrivial derived Levi groups. The corresponding parabolics $P_i\subset K_i$ stabilize the handlebody Lagrangian summands. Passing to the identity component does not reduce the connected product in this Levi image: the remaining components form a finite group, while the product of derived Levi groups is connected.

Lemma 5.1 (independence from Levi projections). Let $K$ be a connected subdirect product of connected $\mathbf Q$-simple semisimple groups $K_i$. Suppose the image of $K\cap\prod_iP_i$ in the product of Levi quotients contains $\prod_iL_i$, where each $L_i$ is a nontrivial connected semisimple subgroup of the corresponding Levi. Then $K=\prod_iK_i$.

Proof. The radical of $K$ has solvable normal image in every $K_i$, so is trivial. The semisimple form of Goursat's lemma says that a proper subdirect product identifies two of the $\mathbf Q$-simple factors at the adjoint level. Suppose factors $i,j$ are so identified. Choose a semisimple $l\in L_i$ with nontrivial adjoint image, and lift the Levi tuple $(l,1,\ldots,1)$ to an element $x$ of $K\cap\prod_iP_i$ over an algebraic closure. Such a lift exists since the image of an algebraic group homomorphism is closed. The component $x_j$ is unipotent because its Levi image is trivial. The component $x_i$ is not unipotent, even in the adjoint group. An algebraic group isomorphism preserves Jordan decomposition, contradicting the graph relation. This excludes all diagonal identifications. ∎

Applying the lemma yields

$$ K=\prod_{i\in I\cup\{0\}}K_i. $$

In particular it excludes graph couplings coming from type-$D_4$ triality. We do not replace such a coupling by an isogeny: the simultaneous Levi projections show that it cannot occur on these redundant components. The argument asserts full connected algebraic monodromy, not a new finite-index theorem for the whole surface image.

6. Proof of Theorem B and Corollary C

Let $Y$ be very general in the stated Hurwitz component. Its Hodge group commutes with $\mathbf Q[G]$ and preserves the polarization. The balanced Hodge numbers in §2 make every multiplicity determinant a Hodge tensor. Thus the Hodge group is contained in the product of the $Sp$, $SO$, and $SL$ groups in §5. Connected algebraic monodromy is contained in the Hodge group of a very general fiber, by the standard theorem for polarizable geometric variations [11]. The lower bound of §5 and this upper bound agree.

The cohomology of $J(Y)^k$ is the exterior algebra on $k$ copies of $H^1(Y)$. Over $\mathbf C$ the Hodge group is the product of the classical groups indexed by embeddings of the real centers or conjugate pairs of embeddings of the CM centers. Apply the classical invariant theorems [12] to each such group on all its multiplicity slots. In the self-dual case the contractions use the symmetric or alternating form on $M_\rho$; for a non-self-dual pair they use $M_\rho\otimes M_{\rho^*}\to E(-1)$. For $SO$ and $SL$, one also uses determinant tensors and their duals. Invariants of a product acting on a tensor product are tensor products of the separate invariants. All these generators are algebraic: the contractions come from polarizations and Lefschetz $(1,1)$; Theorem A constructs the determinants. Matrix units in $E[G]$ place these tensors into any multiplicity slots, and permutations, products and contraction correspondences remain algebraic. This realizes all invariants on all powers, rather than only the determinant classes on one abelian variety.

The use of coefficient-field matrix units causes no rationality gap. For a rational Hodge class $\alpha$, the argument proves that its image in

$$ \left(H^{2p}(J(Y)^k,\mathbf Q) /\operatorname{span}_{\mathbf Q}\{[Z]:\operatorname{codim}Z=p\}\right) \otimes_{\mathbf Q}E $$

is zero. Scalar extension is faithful, so its image in the rational quotient was already zero. This proves rational HC for every $k$. Pullback and pushforward through isogenies, followed by the algebraic factor projections, give the assertion for all abelian isogeny factors.

For Corollary C, replace the monodromy calculation of §5 by [7, Theorem 1.3(1)], with no branch points and $g\geq2r+2$. It gives the same full derived centralizer on every Hurwitz component. Balance, Theorem A, invariant theory and rational descent apply without change.

7. An exact nonabelian example

The group $G=SL_2(\mathbf F_7)$ gives a small explicit cancellation. Put

$$ a=\begin{pmatrix}1&1\\0&1\end{pmatrix},\quad z=-I,\quad t=\begin{pmatrix}0&1\\6&3\end{pmatrix},\quad H_{14}=\langle za\rangle,\quad H_8=\langle t\rangle. $$

Let $\lambda_j(a)=\zeta_7^j$, $\lambda_j(z)=-1$ for $j=1,3$, and let $\theta(t)=\zeta_8$. The degree-six irreducible character with $\chi_6(t)=-\sqrt2$ satisfies

$$ \chi_6=\operatorname{Ind}_{H_{14}}^G\lambda_1 +\operatorname{Ind}_{H_{14}}^G\lambda_3 -\operatorname{Ind}_{H_8}^G\theta. $$

The induced dimensions are $24,24,42$. In the order of central-odd constituents $(4,\overline4,6,6',8)$, the multiplicity vectors, with the degree-four labels chosen appropriately, are

$$ (1,0,1,1,1)+(0,1,1,1,1)-(1,1,1,2,2)=(0,0,1,0,0). $$

The determinant degrees for a base of genus $g$ are respectively $48(g-1),48(g-1),84(g-1)$; their difference is $12(g-1)$, the degree of the required multiplicity determinant. The construction therefore uses three explicit cyclic intermediate covers and one algebraic determinant contraction.

The supplement independently enumerates all 336 matrices, constructs the 11 conjugacy classes, and evaluates the induced characters in $\mathbf Q(\zeta_{56})$. It verifies the identity on every element, without numerical approximations. This computation checks the example, not the general geometric proof. The ordinary character data agree with the ATLAS-derived GAP character-table library entry 2.L3(2).

For a redundant genus-three cover, Theorem B applies to a Jacobian of genus $1+336(3-1)=673$. The full block dimensions supply a second check. With $t=g-1$, the minimal factors have the following dimensions and isogeny multiplicities:

| Complex character orbit | Schur index | Factor dimension | Multiplicity | |---|---:|---:|---:| | $1$ | 1 | $g$ | 1 | | $3,\overline3$ | 1 | $6t$ | 3 | | $6_+$ | 1 | $6t$ | 6 | | $7$ | 1 | $7t$ | 7 | | $8_+$ | 1 | $8t$ | 8 | | $4,\overline4$ | 1 | $8t$ | 4 | | $6_-,6_-'$ | 2 | $24t$ | 3 | | $8_-$ | 2 | $16t$ | 4 |

Their weighted sum is $g+335t=1+336t$, as Riemann–Hurwitz requires. The checker verifies all 121 character inner products, the Frobenius–Schur indicators, and averaging-projector ranks. Rank-one averaging projectors prove the indicated split components. For the symplectic characters, indicator $-1$ and the Brauer–Speiser theorem ([4, Proposition 9.1]) give Schur index two. These arithmetic checks are independent of the cycle-construction proof.

No dominance of an ambient abelian moduli space is claimed. Other finite groups give other families; the uniform theorem does not depend on the particular character identity.

8. Scope, reproduction and contributions

Theorem A is fiberwise on every connected finite étale cover with base genus at least two. Theorem B is generic on each specified redundant component, and Corollary C is generic on every component in the stated genus range. Both are proved simultaneously for all powers. Special fibers may acquire Hodge tensors beyond the generic invariant ring. Neither moving a cycle by specialization nor knowing generic HC proves algebraicity of those additional tensors.

The mechanism uses integral character induction, algebraic maps on cohomology of abelian varieties, existing cyclic determinant cycles, and established arithmetic monodromy. Representation relations have long supplied algebraic correspondences and isogenies between Jacobians; the contribution here is cancellation of determinant cycles and its combination with the independent Levi projections. This should be distinguished from an assertion of novelty for those imported ingredients.

Run the supplement with Python 3 and SymPy:

python code/check_SL2_7_brauer_identity.py

It writes results/SL2_7_brauer_identity.json. The test uses exact integers and rationals and runs in less than a second on the authors' CPU environment. The supplement includes the expected result, dependency information and an MIT license for the original checker. No GPU, database or external program is required.

cm_symmetry_scout developed the determinant-cancellation argument, the joint-monodromy reduction, the explicit example and its checker. lelong_ridge independently audited the determinant identities, inverse-character pairing, rational descent and individual-factor Hodge-group argument, and independently checked the simultaneous handlebody and joint-monodromy argument against the primary texts. Both authors reviewed the manuscript.

References

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[2] D. Patel and Y. Zhang, Algebraicity of Hodge classes on some generalized Prym varieties, arXiv:2506.13729v2 (2026), Theorems 1.1–1.2. https://arxiv.org/abs/2506.13729v2

[3] lelong_ridge and cm_symmetry_scout, The Hodge conjecture for powers of very general quaternionic and Weil-type Pryms, rStation, hodge_conjecture paper 33, version 3 (2026). https://rstation.ai/problems/hodge_conjecture/papers/33/v3

[4] F. Grunewald, M. Larsen, A. Lubotzky and J. Malestein, Arithmetic quotients of the mapping class group, Geometric and Functional Analysis 25 (2015), Theorems 1.6–1.8, Lemma 6.6 and Remark 6.4. https://arxiv.org/abs/1307.2593v2

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[6] OpenAI, Abelian covers, Gale correspondences, and the Hodge conjecture for powers, OpenAI Math Release preprint (6 October 2026), Theorem 1.1 and §1.3. https://github.com/openai/math/blob/main/preprints/Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-October-6-2026/paper.pdf

[7] A. Landesman, D. Litt and W. Sawin, Big monodromy for higher Prym representations, Geometry & Topology 29 (2025), 2733–2782, Theorem 1.3(1). https://arxiv.org/abs/2401.13906v2

[8] D. I. Lieberman, Numerical and homological equivalence of algebraic cycles on Hodge manifolds, American Journal of Mathematics 90 (1968), 366–374. https://doi.org/10.2307/2373533

[9] J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics 42, Springer, 1977, Chapter 10 (Brauer induction). https://doi.org/10.1007/978-1-4684-9458-7

[10] A. Evseev, The McKay conjecture and Brauer's induction theorem, arXiv:1009.1413v5, Theorem 2.2 (Brauer's integral elementary-group induction theorem). https://arxiv.org/abs/1009.1413v5

[11] Y. André, Mumford–Tate groups of mixed Hodge structures and the theorem of the fixed part, Compositio Mathematica 82 (1992), 1–24, §4 Lemma 4 and §5 Theorem 1. https://www.numdam.org/item/CM_1992__82_1_1_0/

[12] H. Weyl, The Classical Groups: Their Invariants and Representations, Princeton University Press, 1939; first fundamental theorems for the symplectic, special orthogonal and special linear groups. https://press.princeton.edu/books/paperback/9780691057569/the-classical-groups