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arxiv: 2604.01975 · v2 · pith:NXGTI226new · submitted 2026-04-02 · 🧮 math.NA · cs.NA· quant-ph

Efficient construction of Lie group-equivariant and permutation-invariant spaces

Elo\"ise Barthelemy , Genevi\`eve Dusson , Camille Hernandez , Liwei Zhang This is my paper

Pith reviewed 2026-05-25 06:17 UTC · model grok-4.3

classification 🧮 math.NA cs.NAquant-ph
keywords Lie groupsequivariant functionspermutation invarianceLie algebranumerical constructionSO(3)SU(2)function approximation
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The pith

A matrix derived from Lie algebra generators has a kernel that exactly spans the space of group-equivariant and permutation-invariant functions of N variables.

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

The paper presents a method to construct functions that remain unchanged under group transformations and permutations of variables by finding the kernel of a matrix built from the group's Lie algebra. This works for any connected linear Lie group and does not require knowing Clebsch-Gordan coefficients in advance. The approach scales linearly with the size of the input space, unlike previous methods that scale exponentially, making it feasible for larger problems. For the rotation group SO(3) and its double cover SU(2), the authors derive the exact dimensions of these spaces from the matrix properties. This allows explicit enforcement of symmetries when approximating functions, which can reduce computational cost especially for moderate numbers of variables.

Core claim

We introduce a practical construction of group-equivariant and permutation-invariant functions of N variables given a finite-dimensional space stable with respect to the group action. The construction applies to any connected linear Lie group and relies on leveraging the Lie algebra to build a matrix M whose kernel is in one-to-one correspondence with the subspace with desired equivariance and invariance properties, removing the need for prior knowledge of Clebsch-Gordan coefficients. A similar construction is proposed for group-equivariant functions alone, without imposing permutation-invariance. For the groups SO(3) and SU(2), we further exploit the structure of the Lie algebra to show the

What carries the argument

The matrix M built from the generators of the Lie algebra, whose kernel is in one-to-one correspondence with the subspace of functions having the desired equivariance and invariance properties.

If this is right

  • The construction applies to arbitrary connected linear Lie groups without case-specific Clebsch-Gordan data.
  • For SO(3) and SU(2) the exact dimension of the space follows from the rank and sparsity pattern of M.
  • The method scales linearly with the dimensionality of the input basis.
  • For large N the dimension of the combined equivariant-invariant space is comparable to that of permutation-invariant spaces alone.
  • Pre-asymptotically the combined space is orders of magnitude smaller than the permutation-invariant space.

Where Pith is reading between the lines

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

  • The linear scaling could make symmetry-constrained approximations practical in high-dimensional settings where exponential methods fail.
  • The same Lie-algebra matrix construction might be adapted to produce bases rather than just dimensions for other groups.
  • Comparing dimensions across groups becomes a uniform linear-algebra task instead of requiring separate representation-theory calculations.

Load-bearing premise

The kernel of the matrix M constructed from the Lie algebra generators is in exact one-to-one correspondence with the desired equivariant and invariant subspace for any connected linear Lie group.

What would settle it

A direct computation of the kernel dimension for a small N, a concrete finite-dimensional stable space, and the group SO(3), compared against the known analytic dimension of the corresponding equivariant permutation-invariant polynomials.

Figures

Figures reproduced from arXiv: 2604.01975 by Camille Hernandez, Elo\"ise Barthelemy, Genevi\`eve Dusson, Liwei Zhang.

Figure 1
Figure 1. Figure 1: Block structure of matrix M2: (a) M2: original derivation; (b) Mf2: Row reordering of the M2 to enable a fast kernel solver; (c) Mup: upper half of Mf2; the final matrix for which we find the kernel. In the plots, the red, green and blue boxes correspond to the matrix in the cases L < Pl, L = Pl, and L > Pl, respectively. All the blocks left blank are zeros. To characterize the kernel of M2, we first switc… view at source ↗
Figure 2
Figure 2. Figure 2: Breakdown of construction time (ms) of the GE-PI bases versus the number of classes in the matrix [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Run time for constructing GE-PI bases using different packages for [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Run time for constructing the GE-PI bases using different packages for correlation orders [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Construction time (ms) of the GE-PI bases versus the number of classes in the matrix [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Relationship between the dimensions of the GE space [PITH_FULL_IMAGE:figures/full_fig_p030_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Exact and estimated dimensions of the GE space [PITH_FULL_IMAGE:figures/full_fig_p031_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Exact and estimated dimensions of the GE-PI space [PITH_FULL_IMAGE:figures/full_fig_p032_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Dimensionality of spaces of functions in [PITH_FULL_IMAGE:figures/full_fig_p032_9.png] view at source ↗
read the original abstract

We introduce a practical construction of group-equivariant and permutation-invariant functions of $N$ variables given a finite-dimensional space stable with respect to the group action. The construction applies to any connected linear Lie group and relies on leveraging the Lie algebra to build a matrix $M$ whose kernel is in one-to-one correspondence with the subspace with desired equivariance and invariance properties, removing the need for prior knowledge of Clebsch--Gordan coefficients. A similar construction is proposed for group-equivariant functions alone, without imposing permutation-invariance. For the groups $SO(3)$ and $SU(2)$, we further exploit the structure of the Lie algebra to demonstrate the sparsity pattern and rank of the matrix $M$, which yields the exact dimension of the group-equivariant and permutation-invariant space, as well as the dimension of the group-equivariant space alone. We demonstrate analytically and verify numerically that the proposed method scales linearly with respect to the dimensionality of the basis, offering a high computational gain compared to existing methods in the literature which typically scale exponentially. We finally perform a dimensionality comparison, showing that for large values of~$N$, the dimension of group-equivariant and permutation-invariant spaces is of comparable order as the dimension of permutation-invariant spaces, while pre-asymptotically, the first dimensionality is orders of magnitude lower than the second. Hence a substantial computational gain can be achieved by explicitly enforcing group-equivariance on top of permutation-invariance when approximating such functions.

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.

Referee Report

2 major / 2 minor

Summary. The paper introduces a construction of Lie group-equivariant and permutation-invariant function spaces of N variables. Given a finite-dimensional space stable under the group action, it builds a matrix M from the infinitesimal generators of the Lie algebra; the kernel of M is asserted to be in one-to-one correspondence with the desired subspace. The method is claimed to apply to any connected linear Lie group and to remove the need for Clebsch-Gordan coefficients. For SO(3) and SU(2) the authors derive the sparsity pattern and rank of M, obtain exact dimensions, prove linear scaling in the basis dimension, and compare dimensions with purely permutation-invariant spaces.

Significance. If the central correspondence holds and the scaling claims are rigorously established, the construction offers a practical, basis-level alternative to coefficient-based methods with linear rather than exponential cost. The analytical sparsity results for the two classical groups and the dimension comparison for large N would be useful in approximation theory and equivariant machine learning.

major comments (2)
  1. [Abstract, paragraph 1] Abstract and the general-construction section: the one-to-one correspondence between ker(M) and the target equivariant-invariant subspace is asserted for arbitrary connected linear Lie groups, yet the manuscript supplies no explicit derivation steps, invocation of the exponential map, or discussion of finite-dimensional representation theory that would confirm the equivalence holds without additional assumptions on the representation or the group.
  2. [SO(3)/SU(2) analysis section] The claim of analytical demonstration of the sparsity pattern and rank of M (which yields the exact dimension) is stated for SO(3) and SU(2), but no derivation or intermediate matrix entries are referenced, preventing verification that the rank calculation is independent of the particular basis chosen for the stable space.
minor comments (2)
  1. [Numerical experiments] The numerical verification of linear scaling is mentioned but no table, figure, or data set (e.g., timing or rank values versus basis dimension) is cited, making the computational-gain statement difficult to assess.
  2. [Section 2] Notation for the matrix M and the stable space should be introduced with a small concrete example (e.g., N=2, SO(3) acting on polynomials of low degree) before the general construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and for recognizing the potential utility of the construction in approximation theory and equivariant machine learning. We address each major comment below and will incorporate the requested clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract, paragraph 1] Abstract and the general-construction section: the one-to-one correspondence between ker(M) and the target equivariant-invariant subspace is asserted for arbitrary connected linear Lie groups, yet the manuscript supplies no explicit derivation steps, invocation of the exponential map, or discussion of finite-dimensional representation theory that would confirm the equivalence holds without additional assumptions on the representation or the group.

    Authors: We agree that the general-construction section would be strengthened by an explicit derivation of the claimed one-to-one correspondence. Although the construction follows from standard facts about the Lie algebra action on finite-dimensional representations of connected linear Lie groups, the manuscript does not spell out the steps involving the exponential map or the relevant representation-theoretic results. We will add a short subsection (or appendix) that invokes the exponential map, recalls the correspondence between Lie-algebra and Lie-group representations for connected groups, and states the precise assumptions under which the kernel of M yields exactly the desired equivariant-invariant subspace. revision: yes

  2. Referee: [SO(3)/SU(2) analysis section] The claim of analytical demonstration of the sparsity pattern and rank of M (which yields the exact dimension) is stated for SO(3) and SU(2), but no derivation or intermediate matrix entries are referenced, preventing verification that the rank calculation is independent of the particular basis chosen for the stable space.

    Authors: We acknowledge that the current text states the sparsity pattern and rank results for SO(3) and SU(2) without displaying the intermediate matrix entries or a self-contained argument that the rank is basis-independent. We will expand the SO(3)/SU(2) analysis section (or add an appendix) to include the explicit computation of the relevant blocks of M for a generic basis of the stable space, together with a short proof that the nullity is invariant under change of basis within that space. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central construction defines matrix M from the infinitesimal generators of the Lie algebra acting on a given finite-dimensional stable space, then takes its kernel as the desired equivariant/invariant subspace. This step rests on the standard theorem that, for finite-dimensional continuous representations of a connected Lie group, invariance under the Lie algebra action is equivalent to invariance under the integrated group action (via the exponential map). The paper implements this equivalence at the basis level without deriving the theorem, without fitting parameters to data, and without load-bearing self-citations that would reduce the claim to prior work by the same authors. No self-definitional loops, fitted inputs renamed as predictions, or ansatzes smuggled via citation appear in the derivation chain. The result is therefore a direct, non-circular computational realization of an externally established fact.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on standard Lie-group representation theory and linear algebra; the only explicit premise is the existence of a finite-dimensional stable space.

axioms (1)
  • domain assumption Given a finite-dimensional space stable with respect to the group action.
    Stated as the starting point for the construction in the abstract.

pith-pipeline@v0.9.0 · 5804 in / 1194 out tokens · 48847 ms · 2026-05-25T06:17:49.068431+00:00 · methodology

discussion (0)

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Reference graph

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5 extracted references · 5 canonical work pages

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