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arxiv: 2604.07850 · v2 · submitted 2026-04-09 · 🧮 math.GR · math.CO

Large products of double cosets for symmetric subgroups

Brendan Pawlowski This is my paper

Pith reviewed 2026-05-10 18:05 UTC · model grok-4.3

classification 🧮 math.GR math.CO
keywords double cosetssymmetric subgroupsLie groupsSU(n)Weyl groupsroot systemsquantum gate decompositioncompact groups
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The pith

Necessary condition on pairs x and y determines when two double cosets multiply to the full Lie group G

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

The paper aims to find which pairs of elements x and y in a simple compact connected Lie group G make the product KxK yK equal the whole group, for a fixed symmetric subgroup K. This matters because the equality tells when two cosets generate all of G, which controls decompositions of group elements into products from the cosets. A necessary condition on x and y is derived that applies to every simply connected G. For the special case G equal to SU(n) the condition becomes a complete classification of the good pairs, except for one family of symmetric subgroups of type AIII. If the results hold, then checking whether two double cosets cover G reduces to verifying a short list of combinatorial conditions instead of examining the entire group.

Core claim

We give a necessary condition on x,y for all simply connected G, and a complete classification when G = SU(n) and any symmetric K ⊆ G except the type AIII case K ≃ S(U(p) × U(n-p)) with p ≠ n/2. We also present some applications of these results to gate decompositions in quantum computing.

What carries the argument

The equality KxK yK = G, analyzed by reducing it through the root system and Weyl group action of the symmetric space to conditions on the conjugacy classes or positions of x and y

If this is right

  • For any simply connected simple compact G one can test the necessary condition on any given x and y without enumerating the whole group.
  • When G is SU(n) the complete list of pairs x and y that work is known for every symmetric K except the unbalanced AIII family.
  • The listed pairs supply explicit decompositions that can be used directly for gate synthesis in quantum computing.
  • The product equals G exactly when x and y satisfy the stated conjugacy or length conditions coming from the Weyl group orbits.

Where Pith is reading between the lines

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

  • The same reduction technique might produce a classification for other classical groups once the case analysis is extended beyond SU(n).
  • In quantum computing the classification restricts which fixed symmetric subgroups can generate all unitaries through two-coset products.
  • The geometric picture of the symmetric space may suggest analogous covering results for non-compact real forms or for products of more than two double cosets.

Load-bearing premise

The structure theory of symmetric spaces is enough to turn the question of whether KxK yK equals G into a finite combinatorial check on root data and Weyl group elements that works case by case for all symmetric subgroups except the excluded AIII family.

What would settle it

An explicit pair x and y in SU(n) for a symmetric K outside the excluded AIII family where the necessary condition holds but KxK yK is strictly smaller than SU(n), or a pair inside the excluded family that violates the claimed classification.

read the original abstract

We consider the problem of classifying pairs $x,y \in G$ such that $K x K y K = G$ where $G$ is a simple compact connected Lie group and $K$ is a symmetric subgroup. We give a necessary condition on $x,y$ for all simply connected $G$, and a complete classification when $G = \operatorname{SU}(n)$ and any symmetric $K \subseteq G$ except the type AIII case $K \simeq \operatorname{S}(\operatorname{U}(p) \times \operatorname{U}(n-p))$ with $p \neq n/2$. We also present some applications of these results to gate decompositions in quantum computing.

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

0 major / 2 minor

Summary. The paper studies pairs x, y in a simple compact connected Lie group G such that the product of double cosets KxK · yK equals G, with K a symmetric subgroup. It establishes a necessary condition on such pairs that holds for every simply connected G, together with an explicit classification of all such pairs when G = SU(n) and K is any symmetric subgroup except the AIII family with p ≠ n/2. Applications to gate decompositions in quantum computing are indicated.

Significance. The necessary condition and the SU(n) classification supply concrete combinatorial criteria, based on root-system and Weyl-group data, for when double-coset products fill the ambient group. This is a standard but technically useful contribution to the structure theory of symmetric spaces; the explicit lists for SU(n) are immediately applicable to questions of generation and decomposition. The quantum-computing applications, if developed, would constitute a concrete interdisciplinary link.

minor comments (2)
  1. [Abstract / Introduction] The abstract states that applications to quantum gate decompositions are presented, yet the introduction or final section should contain at least one explicit example or statement of how the classification is used; this would make the claimed application visible without requiring the reader to reconstruct it.
  2. [§2 or §3] Notation for the symmetric subgroups (especially the AIII family) is introduced in the abstract but should be recalled with a short table or list of root-system data in §2 or §3 so that the case distinctions in the classification are immediately legible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation for minor revision. We address the points in the report below.

read point-by-point responses

  1. Referee: The paper studies pairs x, y in a simple compact connected Lie group G such that the product of double cosets KxK · yK equals G, with K a symmetric subgroup. It establishes a necessary condition on such pairs that holds for every simply connected G, together with an explicit classification of all such pairs when G = SU(n) and K is any symmetric subgroup except the AIII family with p ≠ n/2. Applications to gate decompositions in quantum computing are indicated.

    Authors: We appreciate the referee's accurate summary of our manuscript. It correctly reflects the scope, the necessary condition for simply connected G, the SU(n) classification (with the noted exception), and the indicated applications. revision: no

  2. Referee: The necessary condition and the SU(n) classification supply concrete combinatorial criteria, based on root-system and Weyl-group data, for when double-coset products fill the ambient group. This is a standard but technically useful contribution to the structure theory of symmetric spaces; the explicit lists for SU(n) are immediately applicable to questions of generation and decomposition. The quantum-computing applications, if developed, would constitute a concrete interdisciplinary link.

    Authors: We agree with this evaluation of the combinatorial criteria and their applicability. The quantum-computing applications are indicated in the manuscript but not fully developed, as the primary focus is the classification result. We can expand the relevant section slightly if the editor requests it as part of the minor revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained combinatorial classification

full rationale

The paper derives a necessary condition on pairs (x,y) for general simply connected G and an explicit classification for SU(n) with symmetric K (except one AIII family) by reducing the double-coset product condition KxKyK=G to root-system and Weyl-group combinatorics via standard symmetric-space structure theory. No equations, parameters, or fitted quantities appear; the classification is obtained by direct enumeration of cases rather than any reduction to prior fitted data or self-citations. The exception for the AIII case is stated explicitly, and the argument relies on external, independently verifiable facts about symmetric pairs and Weyl actions, not on any internal self-definition or load-bearing self-reference. This is a standard, self-contained proof in Lie theory with no circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract invokes standard facts about symmetric subgroups of compact Lie groups and the structure of their double cosets; no free parameters, ad-hoc axioms, or new invented entities are introduced.

pith-pipeline@v0.9.0 · 5400 in / 1231 out tokens · 33902 ms · 2026-05-10T18:05:56.198822+00:00 · methodology

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Reference graph

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