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arxiv: 2606.10761 · v1 · pith:P7SF64VDnew · submitted 2026-06-09 · 🧮 math.DG · math.GR· math.GT

Generalizing Lusztig's total positivity II : geometric properties

Olivier Guichard (UNISTRA , IRMA) , Anna Wienhard This is my paper

Pith reviewed 2026-06-27 12:03 UTC · model grok-4.3

classification 🧮 math.DG math.GRmath.GT
keywords positive semigroupflag varietiesnon-negative partspositive structuressymplectic flag varietiestotal positivityLie groupstopology
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The pith

Lie groups admitting positive structures have positive semigroups that induce positive and non-negative parts of flag varieties with determinable topology.

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

The paper establishes key geometric properties of elements belonging to the positive semigroup in a real semisimple Lie group G that admits a positive structure with respect to a subset of positive roots. It introduces the corresponding positive and non-negative parts of flag varieties associated to G. The topology of the non-negative parts is determined in many cases. Explicit descriptions of both the positive and non-negative parts are supplied when the flag variety is symplectic. These results build on the classification and properties of positive structures obtained in the preceding paper of the series.

Core claim

In Lie groups G that admit a positive structure, the positive semigroup has key geometric properties. This structure permits the definition of positive and non-negative parts of flag varieties. The topology of the non-negative parts is determined in many cases. For symplectic flag varieties, explicit descriptions of the positive and non-negative parts are provided.

What carries the argument

The positive semigroup in G associated to a positive structure with respect to a subset Θ of positive roots, which carries the geometric properties used to define positive and non-negative parts of flag varieties.

If this is right

  • The non-negative parts of flag varieties have a topology that can be determined in many cases.
  • Symplectic flag varieties admit explicit descriptions of their positive and non-negative parts.
  • Elements of the positive semigroup satisfy geometric properties that extend the notion of total positivity beyond split real Lie groups.

Where Pith is reading between the lines

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

  • The results supply a geometric realization for the groups classified in the first paper of the series.
  • The explicit symplectic descriptions provide a test case where concrete invariants of the non-negative parts can be calculated directly.
  • The determined topologies indicate that the non-negative parts behave like semialgebraic sets with controlled boundary structure.

Load-bearing premise

That Lie groups admitting positive structures exist and that the positive semigroup inherits the key properties established for them in the preceding paper of the series.

What would settle it

An explicit computation in a concrete Lie group with a positive structure showing that the non-negative part of one of its flag varieties has a topology different from the one determined by the paper, such as a different number of connected components or a different homotopy type.

read the original abstract

Positive structures in Lie groups with respect to a subset $\Theta$ of the set of positive roots provide a generalization of Lusztig's total positivity in split real Lie groups to the setting of general real semisimple Lie groups. In [GW25] Lie groups G admitting a positive structure were classified and many key properties of the unipotent positive semigroups were established. In this article we focus on the positive semigroup in G. We establish key geometric properties of elements in the positive semigroup. Further we introduce corresponding positive and non-negative parts of flag varieties and determine the topology of the non-negative parts of flag varieties in many cases. For symplectic flag varieties we provide explicit descriptions of the positive and non-negative flag varieties.

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 manuscript is the second paper in a series generalizing Lusztig's total positivity from split real Lie groups to general real semisimple Lie groups via positive structures with respect to a subset Θ of positive roots. Building on the classification of groups admitting such structures and the properties of unipotent positive semigroups established in the predecessor [GW25], the paper establishes key geometric properties of elements in the positive semigroup in G. It further introduces the corresponding positive and non-negative parts of flag varieties, determines the topology of the non-negative parts in many cases, and supplies explicit descriptions of the positive and non-negative flag varieties in the symplectic setting.

Significance. If the results hold, the work supplies concrete geometric and topological information about positive structures and their associated flag varieties, extending classical total positivity in a systematic way. The explicit descriptions for symplectic flag varieties constitute a tangible advance that may be useful for further study of homogeneous spaces and related representation-theoretic questions. The manuscript follows the standard architecture of a sequel by openly conditioning new claims on the prior classification while deriving fresh topological and descriptive results.

minor comments (2)
  1. [Abstract] Abstract: the statement that the topology is determined 'in many cases' is imprecise; replace with a reference to the specific theorem(s) or a brief indication of the cases covered.
  2. [Introduction] Introduction: the separation between results imported from [GW25] and the new geometric claims of the present paper should be stated more explicitly (e.g., by a short 'new contributions' paragraph) to assist readers who have not yet consulted the predecessor.

Simulated Author's Rebuttal

0 responses · 0 unresolved

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

Circularity Check

0 steps flagged

No significant circularity; sequel derives new geometric results from stated prior foundation

full rationale

The paper is explicitly Part II and opens by citing [GW25] for the classification of Lie groups admitting positive structures and key properties of unipotent positive semigroups. It then states its own contributions as establishing geometric properties of the positive semigroup in G, introducing positive/non-negative parts of flag varieties, determining the topology of the non-negative parts in many cases, and giving explicit descriptions for symplectic flag varieties. These steps are presented as fresh derivations that take the prior classification as given input. No equation, definition, or claim inside the paper reduces a stated result to its own inputs by construction, renames a fitted quantity as a prediction, or invokes a uniqueness theorem whose only support is an overlapping-author citation that itself lacks independent verification. Self-citation occurs but is not load-bearing for any internal derivation; the new topological and explicit results retain independent mathematical content once the prior classification is granted. This is the normal architecture of a multi-paper series and does not trigger any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the classification of positive structures from the prior paper [GW25] and on standard background facts from Lie group theory. No free parameters, new invented entities, or ad-hoc axioms are visible from the abstract.

axioms (2)
  • domain assumption Existence and properties of positive structures on Lie groups G as classified in [GW25]
    The abstract explicitly states that the work builds on the classification and properties established in the prior paper.
  • standard math Standard axioms and properties of real semisimple Lie groups and their flag varieties
    Background knowledge invoked throughout the field; the abstract assumes these without restatement.

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