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arxiv: 2606.25516 · v1 · pith:BPVTYT64new · submitted 2026-06-24 · 🧮 math.RT · math.AG· math.CO

Total positivity and symmetric spaces

Huanchen Bao This is my paper

Pith reviewed 2026-06-25 20:16 UTC · model grok-4.3

classification 🧮 math.RT math.AGmath.CO
keywords total positivitysymmetric spacedouble Bruhat cellscell decompositionpositive parametrizationsHausdorff closuresubtraction-free mapstotally nonnegative part
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The pith

The totally nonnegative symmetric space G/K admits an explicit cell decomposition with two families of positive parametrizations whose transitions are subtraction-free.

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

The paper defines total positivity on the symmetric space G/K by taking the Hausdorff closure of the image of Lusztig's totally positive part G>0. It introduces double Bruhat cells adapted to this symmetric space and isolates their totally positive pieces. The main results are a cell decomposition of the resulting totally nonnegative space, explicit positive parametrizations for every cell, the closure relations among those cells, and the fact that the natural change-of-coordinates maps between the two parametrization families involve only subtraction-free rational expressions.

Core claim

We define a notion of total positivity for the symmetric space G/K by taking the Hausdorff closure of the image of Lusztig's totally positive part G>0 in G/K. We introduce double Bruhat cells for the symmetric space and define their totally positive pieces. We prove a cell decomposition of the totally nonnegative symmetric space, give explicit positive parametrizations of all cells, establish closure relations, and show that the transition maps between the two natural families of parametrizations are subtraction-free.

What carries the argument

The totally nonnegative symmetric space, obtained as the Hausdorff closure of the image of G>0 inside G/K, together with its double Bruhat cells and their totally positive pieces.

If this is right

  • The cells form a stratification of the totally nonnegative symmetric space.
  • Every cell carries two families of explicit positive parametrizations.
  • Closure relations among cells are completely determined by the parametrizations.
  • All transition maps between the two families are subtraction-free rational maps.

Where Pith is reading between the lines

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

  • The subtraction-free transitions make the structure compatible with semiring or tropical operations on the same coordinate charts.
  • The construction supplies a combinatorial model that may be used to study positive representations or canonical bases attached to the symmetric space.

Load-bearing premise

The Hausdorff closure of the image of G>0 inside G/K yields a well-behaved totally nonnegative part whose cells admit the stated positive parametrizations and closure relations.

What would settle it

A concrete point in the Hausdorff closure whose local coordinates in one of the two natural parametrizations require a subtraction, or a pair of cells whose closure relation fails to match the predicted combinatorial pattern.

read the original abstract

We define a notion of total positivity for the symmetric space $G/K$ by taking the Hausdorff closure of the image of Lusztig's totally positive part $G_{>0}$ in $G/K$. We introduce double Bruhat cells for the symmetric space and define their totally positive pieces. We prove a cell decomposition of the totally nonnegative symmetric space, give explicit positive parametrizations of all cells, establish closure relations, and show that the transition maps between the two natural families of parametrizations are subtraction-free.

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 / 1 minor

Summary. The paper defines total positivity for the symmetric space G/K as the Hausdorff closure of the image of Lusztig's totally positive part G>0. It introduces double Bruhat cells for the symmetric space together with their totally positive pieces. The central results are a cell decomposition of the totally nonnegative symmetric space, explicit positive parametrizations of all cells, closure relations among the cells, and the claim that transition maps between the two natural families of parametrizations are subtraction-free.

Significance. If the stated theorems hold, the work supplies a Lusztig-style positive structure on symmetric spaces, furnishing combinatorial parametrizations and closure data that are likely to be useful in the study of positive representations, canonical bases, and geometric structures on G/K. The subtraction-free transition property is a concrete algebraic strength that aligns with existing total-positivity literature.

minor comments (1)
  1. The abstract asserts that proofs of the cell decomposition, parametrizations, closure relations, and subtraction-free transitions exist, but the provided text does not display the actual derivations; a referee therefore cannot yet verify the absence of gaps in the topological or algebraic arguments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for recognizing the potential utility of a Lusztig-style positive structure on symmetric spaces, including the combinatorial parametrizations, closure relations, and subtraction-free transitions. The recommendation is listed as uncertain, yet no specific major comments or points of concern are provided in the report. Accordingly, we have no revisions to propose and maintain that the stated theorems hold as written.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines the totally nonnegative symmetric space as the Hausdorff closure of the image of G>0 in G/K, then introduces double Bruhat cells and proves cell decomposition, explicit positive parametrizations, closure relations, and subtraction-free transition maps. These are established via direct constructions following Lusztig-style total positivity methods, with no reduction of results to fitted parameters, self-definitional equations, or load-bearing self-citations. All claims rest on independent mathematical arguments from the given definitions rather than circular reuse of outputs as inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

Based solely on the abstract, the paper relies on standard background from algebraic groups and topology; no free parameters or invented entities beyond the new definitions are visible.

axioms (3)
  • domain assumption G is a connected reductive algebraic group over an algebraically closed field of characteristic zero
    Standard setup for Lusztig total positivity and symmetric spaces G/K
  • domain assumption K is the fixed-point subgroup of an involution of G
    Definition of the symmetric space
  • standard math The quotient G/K carries the quotient topology in which Hausdorff closure is taken
    Used to define the totally nonnegative part
invented entities (1)
  • totally nonnegative symmetric space no independent evidence
    purpose: Hausdorff closure of the image of G_{>0} in G/K
    Newly defined object whose properties are proved in the paper

pith-pipeline@v0.9.1-grok · 5598 in / 1519 out tokens · 18354 ms · 2026-06-25T20:16:33.357599+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

23 extracted references · 7 canonical work pages

  1. [1]

    Huanchen Bao and Xuhua He,A Birkhoff–Bruhat atlas for partial flag varieties, Indag. Math. (N.S.)32(2021), no. 5, 1152–1173

    2021

  2. [2]

    8, 2037– 2069, DOI 10.2140/ant.2021.15.2037

    ,Flag manifolds over semifields, Algebra Number Theory15(2021), no. 8, 2037– 2069, DOI 10.2140/ant.2021.15.2037

    work page doi:10.2140/ant.2021.15.2037 2021

  3. [3]

    ,Total positivity in twisted product of flag varieties(2022), available atarXiv: 2211.11168

    arXiv 2022

  4. [4]

    Math.237(2024), no

    ,Product structure and regularity theorem for totally nonnegative flag varieties, In- vent. Math.237(2024), no. 1, 1–47, DOI 10.1007/s00222-024-01256-2

    work page doi:10.1007/s00222-024-01256-2 2024

  5. [5]

    Huanchen Bao and Jinfeng Song,Coordinate rings on symmetric spaces(2024), available atarXiv:2402.08258

    arXiv 2024

  6. [6]

    ,Symmetric subgroup schemes, Frobenius splittings, and quantum symmetric pairs (2022), available atarXiv:2212.13426

    arXiv 2022

  7. [7]

    Huanchen Bao and Weiqiang Wang,Canonical bases arising from quantum symmetric pairs (2018), available atarXiv:1610.09271

    Pith/arXiv arXiv 2018

  8. [8]

    G-actions with close orbit spaces

    Kei Yuen Chan, Jiang-Hua Lu, and Simon Kai-Ming To,On intersections of conjugacy classes and Bruhat cells, Transform. Groups15(2010), no. 2, 243–260, DOI 10.1007/s00031- 010-9084-7. MR2657442

    work page doi:10.1007/s00031- 2010

  9. [9]

    996, Springer, Berlin, 1983, pp

    Corrado De Concini and Claudio Procesi,Complete symmetric varieties, Invariant theory (Montecatini, 1982), Lecture Notes in Math., vol. 996, Springer, Berlin, 1983, pp. 1–44

    1982

  10. [10]

    Sam Evens and Jiang-Hua Lu,On the variety of Lagrangian subalgebras. II, Ann. Sci. ´Ecole Norm. Sup. (4)39(2006), no. 2, 347–379

    2006

  11. [11]

    Karp, and Thomas Lam

    Pavel Galashin, Steven N. Karp, and Thomas Lam,Regularity theorem for totally nonneg- ative flag varieties, J. Amer. Math. Soc.35(2022), no. 2, 513–579, DOI 10.1090/jams/983

    work page doi:10.1090/jams/983 2022

  12. [12]

    Jiang-Hua Lu,On theT-leaves and the ranks of a Poisson structure on twisted conjugacy classes, Indag. Math. (N.S.)25(2014), no. 5, 1102–1121, DOI 10.1016/j.indag.2014.07.011

    work page doi:10.1016/j.indag.2014.07.011 2014

  13. [13]

    George Lusztig,Total positivity in reductive groups, Lie theory and geometry (1994), 531– 568

    1994

  14. [14]

    ,Introduction to total positivity, Positivity in Lie theory: open problems, 1998, pp. 133–145

    1998

  15. [15]

    Theory26(2022), 1025–1046, DOI 10.1090/ert/628, available atarXiv:2107.13447

    ,Total positivity in symmetric spaces, Represent. Theory26(2022), 1025–1046, DOI 10.1090/ert/628, available atarXiv:2107.13447

    work page doi:10.1090/ert/628 2022

  16. [16]

    R. J. Marsh and K. Rietsch,Parametrizations of flag varieties(2004), available atarXiv: math/0307017

    arXiv 2004

  17. [17]

    R. W. Richardson,Intersections of double cosets in algebraic groups, Indag. Math. (N.S.)3 (1992), no. 1, 69–77, DOI 10.1016/0019-3577(92)90028-J. MR1157520

    work page doi:10.1016/0019-3577(92)90028-j 1992

  18. [18]

    R. W. Richardson and T. A. Springer,The Bruhat order on symmetric varieties, Geom. Dedicata35(1990), 389–436

    1990

  19. [19]

    Rietsch,Closure relations for totally nonnegative cells inG/P(2005), available atarXiv: math/0509137

    K. Rietsch,Closure relations for totally nonnegative cells inG/P(2005), available atarXiv: math/0509137

    Pith/arXiv arXiv 2005

  20. [20]

    Jinfeng Song,Symmetric subgroup schemes(2025), available atarXiv:2512.01398

    arXiv 2025

  21. [21]

    T. A. Springer,Linear algebraic groups, 2nd ed., Birkh¨ auser, 1998

    1998

  22. [22]

    ,Some results on algebraic groups with involutions, Algebraic groups and related topics, 1985, pp. 525–543

    1985

  23. [23]

    Wikipedia,https://en.wikipedia.org/wiki/ Satake_diagram

    Wikipedia contributors,Satake diagram. Wikipedia,https://en.wikipedia.org/wiki/ Satake_diagram. Accessed 2026-06-07. Department of Mathematics, National University of Singapore, Singapore. Email address:huanchen@nus.edu.sg

    2026