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arxiv: 2604.18894 · v1 · submitted 2026-04-20 · 🧮 math.RT · math.CO· math.OC

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Kazhdan-Lusztig Basis and Optimization

Geordie Williamson, Tom Goertzen

Pith reviewed 2026-05-10 02:42 UTC · model grok-4.3

classification 🧮 math.RT math.COmath.OC
keywords Kazhdan-Lusztig basisSpecht modulesHecke algebraquadratic optimizationcanonical basesYoung's seminormal basisinvariant conesGelfand-Tsetlin basis
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The pith

A quadratic optimization problem over unitriangular bases with non-negative action of 1+s recovers the Kazhdan-Lusztig basis as the unique maximizer for Specht modules of all partitions up to size 7.

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

The paper develops a conjectural method to locate canonical bases of the Hecke algebra at q=1 by solving continuous quadratic optimization problems. It considers bases in Specht modules that remain unitriangular to the polytabloid basis while satisfying the constraint that every operator 1+s acts non-negatively. Unique minimal and maximal cones invariant under all such operators are shown to exist, and the Kazhdan-Lusztig basis is proved to span the maximal cone for hook shapes, two-column shapes, and partitions of the form (n-2,2). Minimization of the trace of the Gram matrix recovers Young's seminormal basis exactly, while maximization recovers the Kazhdan-Lusztig basis for every partition of n at most 7. The same framework applied to irreducible representations of sl_n shows that the Gelfand-Tsetlin basis is the unique minimizer, with the canonical basis conjectured to be the maximizer in small ranks. A sympathetic reader would care because the method offers an algebraic and computational route to canonical bases that does not presuppose their combinatorial construction.

Core claim

By defining an optimization problem over bases that are unitriangular with respect to the polytabloid basis and subject to the constraint that the operators 1+s act non-negatively for every simple reflection s, the feasible region is proved to be a compact semialgebraic set that corresponds to a hierarchy of cones invariant under all 1+s. Minimizing the trace of the Gram matrix uniquely recovers Young's seminormal basis. Maximization computationally recovers the Kazhdan-Lusztig basis uniquely for every partition of n at most 7. For hook shapes, two-column shapes, and partitions (n-2,2) the Kazhdan-Lusztig basis is proved to span the maximal invariant cone. The framework extends to sl_n, with

What carries the argument

The compact semialgebraic set of unitriangular bases on which every 1+s acts non-negatively, interpreted as a hierarchy of invariant cones inside Specht modules.

If this is right

  • For hook shapes, two-column shapes, and partitions of the form (n-2,2), the Kazhdan-Lusztig basis spans the maximal cone invariant under all 1+s.
  • Minimizing the trace of the Gram matrix over the feasible region uniquely recovers Young's seminormal basis.
  • Maximization over the same region computationally recovers the Kazhdan-Lusztig basis for every partition of n at most 7.
  • In higher ranks the optimization detects deviations from the Kazhdan-Lusztig basis and may select other natural positive bases such as the Springer basis or p-canonical bases.
  • For irreducible representations of sl_n the Gelfand-Tsetlin basis is the unique minimizer, and the canonical basis is conjectured to be the maximizer in small ranks.

Where Pith is reading between the lines

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

  • If the maximizer continues to select the Kazhdan-Lusztig basis for some but not all partitions beyond n=7, the objective function may serve as a numerical test distinguishing the Kazhdan-Lusztig basis from other positive bases.
  • The hierarchy of invariant cones defined by the feasible region offers a geometric language for comparing different positive bases inside the same module.
  • The same quadratic program could be run on modules for other Hecke algebras or quantized enveloping algebras where unitriangularity and positivity constraints are available.

Load-bearing premise

The positivity constraint that 1+s acts non-negatively together with unitriangularity to the polytabloid basis is sufficient to isolate the Kazhdan-Lusztig basis as the unique maximizer of the chosen quadratic objective rather than some other positive basis.

What would settle it

Compute the maximizer for a concrete partition of 8 such as (6,2) or (5,3) and check whether the resulting basis vectors coincide exactly with the known Kazhdan-Lusztig basis or instead select a different positive basis such as the Springer basis.

Figures

Figures reproduced from arXiv: 2604.18894 by Geordie Williamson, Tom Goertzen.

Figure 1
Figure 1. Figure 1: Bruhat graph in type A2 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: W-graph Γ (3,2): All edge weights are equal to 1. For the correspondence to the vertex labelling via descents of standard tableaux, see [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Optimal solution in the case where W is of type I2(6) and I2(10) in Problem 3.1 when dropping condition (4) in Problem 3.1. Blocks of the same form are highlighted in the same colour. We can simplify condition 3 in Problem 3.1 by imposing vanishing conditions on the structure constants, for instance µ z x,y = 0 if ℓ(x) + ℓ(y) < ℓ(z), which we know hold for the Kazhdan–Lusztig basis. However, in general the… view at source ↗
Figure 4
Figure 4. Figure 4: Strategy for showing that the dual Kazhdan–Lusztig left cell basis spans the minimal 1 + s, s ∈ S invariant cone inside S λ , for λ = (5, 2). Tableaux that are maximal with regard to the complement of their descent set are marked in red. The example suggests the following strategy for verifying the minimal invariant cone property for the dual Kazhdan–Lusztig left cell representation with basis elements of … view at source ↗
Figure 5
Figure 5. Figure 5: Invariant cones for x ∈ [−1, −0.5] under operators 1+s1, 1+s2 for the Specht module S (2,1). The minimal cone given by the fundamental weight basis is also highlighted. The maximal cone is attained for x = −1 and agrees with the Kazhdan–Lusztig basis. For x = −0.5, we obtain Young’s seminormal basis. We can generalize the observations of the example in the following theorem. Theorem 5.10. The minimum Amin … view at source ↗
read the original abstract

We describe a conjectural approach to obtaining canonical bases of the Hecke algebra at $q=1$ via continuous quadratic optimization. We focus on Specht modules $S^\lambda$ and proper cones inside $S^\lambda$ that are invariant under the action of $1+s$ for all simple reflections $s\in S$. We show that there are unique minimal and maximal cones invariant under all $1+s$. For hook shapes, two-column shapes, and partitions of the form $(n-2,2)$, we prove that the Kazhdan--Lusztig basis spans this maximal cone. More generally, we define an optimization problem over bases that are unitriangular with respect to the polytabloid basis, subject to the constraint that the operators $1+s$ act non-negatively. We prove that the feasible region forms a compact semialgebraic set, and interpret it in terms of a hierarchy of invariant cones under all $1+s$. We demonstrate that minimizing the trace of the Gram matrix uniquely recovers Young's seminormal basis. Furthermore, we verify computationally that maximization uniquely recovers the Kazhdan--Lusztig basis for all partitions of $n\leq 7$. In higher ranks, the optimization detects deviations from the Kazhdan--Lusztig basis and may favour other natural positive bases, such as the Springer basis or $p$-canonical bases. Finally, we extend this framework to irreducible representations of $\mathfrak{sl}_n$. We observe that the Gelfand--Tsetlin basis corresponds to the unique minimizer, and we conjecture that the canonical basis corresponds to the maximum in small ranks.

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 manuscript proposes a conjectural optimization framework for characterizing the Kazhdan-Lusztig basis of the Hecke algebra at q=1 within Specht modules S^λ. It proves the existence of unique minimal and maximal cones invariant under the action of all 1+s (s simple reflection), shows that the KL basis spans the maximal cone for hook shapes, two-column shapes, and partitions (n-2,2), proves that minimizing the trace of the Gram matrix recovers Young's seminormal basis, and supplies computational verification that maximization of the chosen quadratic objective uniquely recovers the KL basis for all partitions of n≤7. The approach is extended to irreducible representations of sl_n, where the Gelfand-Tsetlin basis is observed to be the unique minimizer and the canonical basis is conjectured to be the maximizer in small ranks.

Significance. If the conjectural uniqueness of the maximizer holds beyond the verified range, the work supplies a new continuous, optimization-based characterization of canonical bases that links positivity cones, unitriangularity, and quadratic objectives. The proven results for special shapes, the recovery of the seminormal basis as minimizer, and the explicit semialgebraic description of the feasible region constitute concrete advances. The computational evidence up to n=7 together with the detection of other positive bases (Springer, p-canonical) in higher rank provides useful empirical data and highlights dimension-dependent phenomena.

major comments (2)
  1. [Optimization problem and computational verification] The manuscript proves that the feasible region is a compact semialgebraic set and that the KL basis spans the maximal cone for the listed special shapes, but the central claim that maximization uniquely isolates the KL basis (rather than another positive basis) rests on numerical verification only up to n=7. No algebraic argument is supplied showing that the chosen quadratic objective is strictly larger at the KL point than at any other feasible unitriangular matrix satisfying the 1+s non-negativity constraints.
  2. [Extension to sl_n] In the extension to sl_n representations, the observation that the Gelfand-Tsetlin basis is the unique minimizer is stated without an accompanying proof or reference to a prior result, while the conjecture that the canonical basis is the maximizer lacks even the small-rank computational support provided for the Hecke-algebra case.
minor comments (2)
  1. [Abstract] The abstract and introduction should explicitly list the partitions or shapes for which the KL-maximizer claim is proven versus those verified only computationally.
  2. [Definition of the optimization problem] Clarify the precise quadratic objective function used for maximization (as opposed to the trace minimization for the seminormal basis) so that the optimization problem can be reproduced without ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive evaluation, and constructive suggestions. We address each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: The manuscript proves that the feasible region is a compact semialgebraic set and that the KL basis spans the maximal cone for the listed special shapes, but the central claim that maximization uniquely isolates the KL basis (rather than another positive basis) rests on numerical verification only up to n=7. No algebraic argument is supplied showing that the chosen quadratic objective is strictly larger at the KL point than at any other feasible unitriangular matrix satisfying the 1+s non-negativity constraints.

    Authors: We agree that the uniqueness of the Kazhdan-Lusztig basis as the maximizer of the quadratic objective is supported solely by computational verification up to n≤7, without a general algebraic proof that the objective is strictly larger at the KL point than at other feasible unitriangular bases satisfying the non-negativity constraints. This is a genuine limitation of the present work; we prove the KL basis spans the maximal cone only for the listed special shapes, and the general case remains conjectural. In the revised manuscript we will state this distinction more explicitly in the abstract, introduction, and conclusion, while retaining the computational evidence as supporting data for the conjecture. revision: partial

  2. Referee: In the extension to sl_n representations, the observation that the Gelfand-Tsetlin basis is the unique minimizer is stated without an accompanying proof or reference to a prior result, while the conjecture that the canonical basis is the maximizer lacks even the small-rank computational support provided for the Hecke-algebra case.

    Authors: The identification of the Gelfand-Tsetlin basis as unique minimizer in the sl_n setting is indeed an observational claim based on explicit low-rank computations rather than a general proof. We will revise the text to clarify this and, where relevant, add a reference to related literature on Gelfand-Tsetlin bases. For the conjecture that the canonical basis is the maximizer, we will include the corresponding small-rank computational verification to match the evidentiary standard used for the Hecke-algebra case. These clarifications and additions will appear in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; constraints and verification are independent of target basis

full rationale

The paper defines the feasible region explicitly via unitriangularity to the polytabloid basis plus non-negativity of all 1+s operators, without embedding the Kazhdan-Lusztig basis into the definition. It proves existence of unique min/max invariant cones and that the KL basis spans the maximal cone for hooks, two-column shapes, and (n-2,2). Minimization is shown algebraically to recover the seminormal basis. The claim that maximization recovers KL for n≤7 is presented as computational output on the independently defined semialgebraic set, not as an input or self-definition. No self-citation is load-bearing for the central statements, and no ansatz or renaming reduces the result to its own premises by construction. The framework remains self-contained against external algebraic definitions of the bases.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the standard axioms of the Hecke algebra at q=1, the definition of Specht modules, and the existence of the polytabloid basis; the positivity constraints and the choice of quadratic objective are introduced by the paper.

axioms (2)
  • standard math Specht modules admit a polytabloid basis with respect to which other bases of interest are unitriangular.
    Invoked when the optimization is defined over unitriangular bases.
  • domain assumption The operators 1+s for simple reflections s act on the module and the non-negativity condition is well-defined.
    Central to the definition of the feasible region.

pith-pipeline@v0.9.0 · 5597 in / 1499 out tokens · 44291 ms · 2026-05-10T02:42:03.727227+00:00 · methodology

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