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arxiv: 1907.06755 · v1 · pith:Z4HWASKPnew · submitted 2019-07-15 · 🧮 math.GR

Finite Singular Orbit Modules for Algebraic Groups

Aluna Rizzoli This is my paper

Pith reviewed 2026-05-24 20:52 UTC · model grok-4.3

classification 🧮 math.GR
keywords algebraic groupsirreducible modulesfinite orbitssingular 1-spacesorthogonal modulesdouble cosetsparabolic subgroupsspin modules
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The pith

All faithful irreducible orthogonal modules with finitely many orbits on singular 1-spaces are determined for simple and maximal-semisimple algebraic groups.

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

The paper extends a prior classification of modules having finitely many orbits on all subspaces to the restricted setting of singular 1-spaces inside orthogonal modules. It produces a complete list of every faithful irreducible module over simple or maximal-semisimple connected algebraic groups that meets the finite-orbit condition on those singular lines. The result directly settles the double-coset finiteness question for a classical group and one of its maximal parabolic subgroups. The list contains both previously known cases and new examples, including a five-dimensional module for SL_2 and the spin module for B_6 in characteristic two.

Core claim

We determine all faithful irreducible modules for simple and maximal-semisimple connected algebraic groups that are orthogonal and have finitely many orbits on singular 1-spaces, building on the classification of modules with finitely many orbits on subspaces. This provides a solution to the double coset problem when one subgroup is a maximal parabolic in a classical group SO_n. The resulting examples range from a 5-dimensional module for SL_2 to the spin module for B_6 in characteristic 2.

What carries the argument

The prior classification of modules with finitely many orbits on subspaces, extended by restricting attention to orbits on singular 1-spaces.

If this is right

  • The pairs consisting of a classical group SO_n and its maximal parabolic subgroup P_1 have finitely many double cosets precisely when the corresponding module appears in the list.
  • The five-dimensional module for SL_2 and the spin module for B_6 in characteristic two each satisfy the finite-orbit condition on singular 1-spaces.
  • The determination covers every simple and every maximal-semisimple connected algebraic group.

Where Pith is reading between the lines

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

  • The same orbit-counting technique may extend to non-orthogonal modules or to questions about orbits on higher-dimensional singular subspaces.
  • The double-coset result for maximal parabolics could be compared with known finiteness criteria for other parabolic subgroups in the same groups.
  • Characteristic-two phenomena in the B_6 spin module suggest checking whether similar modules exist for other exceptional groups in small characteristics.

Load-bearing premise

The prior classification of modules with finitely many orbits on subspaces is both complete and directly applicable to the new singular 1-space case.

What would settle it

An orthogonal faithful irreducible module over a simple algebraic group that is absent from the listed examples yet still possesses only finitely many orbits on singular 1-spaces would falsify the classification.

read the original abstract

Building on the classification of modules for algebraic groups with finitely many orbits on subspaces, we determine all faithful irreducible modules for simple and maximal-semisimple connected algebraic groups that are orthogonal and have finitely many orbits on singular $1$-spaces. This question is naturally connected with the problem of finding for which pairs of subgroups $H,K$ of an algebraic group $G$ there are finitely many $(H,K)$-double cosets. This paper provides a solution to the question when $K$ is a maximal parabolic subgroup $P_1$ of a classical group $SO_n$. We find an interesting range of new examples ranging from a $5$-dimensional module for $SL_2$ to the spin module for $B_6$ in characteristic $2$.

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. Building on the classification of modules for algebraic groups with finitely many orbits on subspaces, the paper determines all faithful irreducible modules for simple and maximal-semisimple connected algebraic groups that are orthogonal and have finitely many orbits on singular 1-spaces. It connects this to the problem of finding pairs of subgroups H,K with finitely many (H,K)-double cosets when K is a maximal parabolic P1 of a classical group SO_n, and exhibits new examples ranging from a 5-dimensional module for SL_2 to the spin module for B_6 in characteristic 2.

Significance. If the classification holds, the result refines the theory of orbit-finite representations by isolating the orthogonal/singular case, yielding concrete new examples across characteristics and group types. The systematic reduction to the prior subspace-orbit classification, combined with explicit case analysis on simple and maximal-semisimple groups, provides a clear extension with potential applications to double-coset problems. The work credits the completeness of the invoked prior list and supplies falsifiable examples that can be checked directly.

minor comments (2)
  1. The connection between the singular 1-space orbit condition and the prior subspace classification could be stated more explicitly in the introduction to clarify why no additional cases arise.
  2. A summary table listing all new examples (with group, dimension, characteristic, and reference to the prior list) would improve readability of the case analysis.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No circularity: classification extends prior work via case analysis without reduction to inputs

full rationale

The paper determines the modules by reducing to and extending a prior classification of modules with finitely many orbits on subspaces, then performing case analysis on simple and maximal-semisimple groups to isolate orthogonal/singular cases and exhibit explicit new examples (e.g., 5-dimensional SL_2 module, B_6 spin module in char 2). No equations or steps are shown that define a quantity in terms of itself, rename a fitted input as a prediction, or rely on a self-citation chain whose validity is internal to this paper. The derivation remains self-contained against the external benchmark of the invoked prior classification.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no visible free parameters, axioms, or invented entities; the central claim rests on an unstated prior classification whose details are not provided.

pith-pipeline@v0.9.0 · 5643 in / 998 out tokens · 15815 ms · 2026-05-24T20:52:31.640960+00:00 · methodology

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Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    Lower bounds for essential dimensions in characteristic 2 via ortho gonal representations

    Babic, A., and Chernousov, V. Lower bounds for essential dimensions in characteristic 2 via ortho gonal representations. Pacific Journal of Mathematics 279 (2015), 37–63

    work page 2015

  2. [2]

    ´El´ ements unipotents et sous-groupes paraboliques de groupes r´ eductifs

    Borel, A., and Tits, J. ´El´ ements unipotents et sous-groupes paraboliques de groupes r´ eductifs. I. Invent. Math. 12 (1971), 95–104

    work page 1971

  3. [3]

    Double coset density in classical algebraic groups

    Brundan, J. Double coset density in classical algebraic groups. Trans. Amer. Math. Soc. 352 (2000), 1405–1436. 32

    work page 2000

  4. [4]

    M., and Cooperstein, B

    Cohen, A. M., and Cooperstein, B. N. The 2-spaces of the standard E6(q)-module. Geom. Dedicata 25 (1988), 467–480

    work page 1988

  5. [5]

    M., and Helminck, A

    Cohen, A. M., and Helminck, A. G. Trilinear alternating forms on a vector space of dimension 7. Comm. Algebra 16 (1988), 1–25

    work page 1988

  6. [6]

    On the trivectors of a 6-dimensional symplectic vector space

    De Bruyn, B., and Kwiatkowski, M. On the trivectors of a 6-dimensional symplectic vector space. II. Linear Algebra Appl. 437 (2012), 1215–1233

    work page 2012

  7. [7]

    Spinors of 13-dimensional space

    Gatti, V., and Viniberghi, E. Spinors of 13-dimensional space. Adv. in Math. 30 (1978), 137–155

    work page 1978

  8. [8]

    M., Liebeck, M

    Guralnick, R. M., Liebeck, M. W., Macpherson, D., and Seitz, G. M. Modules for algebraic groups with finitely many orbits on subspaces. J. Algebra 196 (1997), 211–250

    work page 1997

  9. [9]

    Humphreys, J. E. Introduction to Lie algebras and representation theory . Graduate Texts in Mathematics, Vol. 9. Springer-Verlag, New York-Berlin, 1972

    work page 1972

  10. [10]

    A classification of spinors up to dimension twelve

    Igusa, J.-i. A classification of spinors up to dimension twelve. Amer. J. Math. 92 (1970), 997–1028

    work page 1970

  11. [11]

    Invariant forms on irreducible modules of simple algebraic groups

    Korhonen, M. Invariant forms on irreducible modules of simple algebraic groups. J. Algebra 480 (2017), 385–422

    work page 2017

  12. [12]

    W., Saxl, J., and Seitz, G

    Liebeck, M. W., Saxl, J., and Seitz, G. M. Factorizations of simple algebraic groups. Trans. Amer. Math. Soc. 348 (1996), 799–822

    work page 1996

  13. [13]

    W., and Seitz, G

    Liebeck, M. W., and Seitz, G. M. On the subgroup structure of classical groups. Invent. Math. 134 (1998), 427–453

    work page 1998

  14. [14]

    W., and Seitz, G

    Liebeck, M. W., and Seitz, G. M. On the subgroup structure of exceptional groups of Lie type. Trans. Amer. Math. Soc. 350 (1998), 3409–3482

    work page 1998

  15. [15]

    W., and Seitz, G

    Liebeck, M. W., and Seitz, G. M. Unipotent and nilpotent classes in simple algebraic groups and Lie algebras, vol. 180 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2012

    work page 2012

  16. [16]

    Small degree representations of finite Chevalley groups in defining c haracteristic

    L¨ubeck, F. Small degree representations of finite Chevalley groups in defining c haracteristic. LMS J. Comput. Math. 4 (2001), 135–169

    work page 2001

  17. [17]

    Linear algebraic groups and finite groups of Lie type , vol

    Malle, G., and Testerman, D. Linear algebraic groups and finite groups of Lie type , vol. 133 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2011

    work page 2011

  18. [18]

    Popov, V. L. Classification of spinors of dimension fourteen. Transactions of the Moscow Mathematical Society 1 (1980), 181–232

    work page 1980

  19. [19]

    Uber trilineare alternierende Formen in sechs und sieben Ve r anderlichen und die durch siedefinierten geometrischen Gebilde

    Reichel, W. Uber trilineare alternierende Formen in sechs und sieben Ve r anderlichen und die durch siedefinierten geometrischen Gebilde. PhD thesis, Greifswald, 1907

    work page 1907

  20. [20]

    Trivecteurs de rang 6

    Revoy, P. Trivecteurs de rang 6. In Colloque sur les formes quadratiques (Montpellier, 1977) , M´ emoires de la Soci´ et´ e Math´ ematique de France. Soci´ et´ e math´ ematique de France, 1979, pp. 141–155

    work page 1977

  21. [21]

    Schouten, J. A. Klassifizierung der alternierenden gr¨ oszen dritten grades in 7 dime nsionen. Rendiconti del Circolo Matematico di Palermo (1884-1940) 55 (Dec 1931), 137–156. 33

    work page 1940

  22. [22]

    Seitz, G. M. Double cosets in algebraic groups. In Algebraic groups and their representations (Cambridge, 1997), vol. 517 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Kluwer Acad. Publ., Dordrecht, 1998, pp. 241–257

    work page 1997

  23. [23]

    A., and Steinberg, R

    Springer, T. A., and Steinberg, R. Conjugacy classes. In Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton , N.J., 1968/69) , Lecture Notes in Mathemat- ics, Vol. 131. Springer, Berlin, 1970, pp. 167–266

    work page 1968

  24. [24]

    Lectures on Chevalley groups , vol

    Steinberg, R. Lectures on Chevalley groups , vol. 66 of University Lecture Series. American Mathematical Society, Providence, RI, 2016

    work page 2016

  25. [25]

    Group theory

    Suzuki, M. Group theory. I, vol. 247 of Grundlehren der Mathematischen Wissenschaften . Springer-Verlag, Berlin-New York, 1982. 34

    work page 1982