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arxiv: 2104.14530 · v2 · submitted 2021-04-29 · 🧮 math.RT · math.CO· math.OA· math.PR

Reflection length with two parameters in the asymptotic representation theory of type B/C and applications

Marek Bo\.zejko , Maciej Do{\l}\k{e}ga , Wiktor Ejsmont , \'Swiatos{\l}aw R. Gal This is my paper

Pith reviewed 2026-05-24 13:24 UTC · model grok-4.3

classification 🧮 math.RT math.COmath.OAmath.PR
keywords hyperoctahedral groupreflection lengthpositive definite functionsextreme charactersSchur-Weyl dualityFock spaceAskey-Wimp-Kerov distributionCoxeter groups type B and D
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The pith

A two-parameter refinement of reflection length on the infinite hyperoctahedral group is positive definite precisely when it equals an extreme character.

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

The paper introduces the function φ_{q+,q-} on the infinite hyperoctahedral group as a signed bivariate version of reflection length that separately tracks long and short reflections. It proves this function is positive definite if and only if it coincides with an extreme character and classifies the exact ranges of q+ and q- for which this holds. The classification produces representations of the group via a two-parameter action of B(n) on an n-fold tensor product, directly generalizing the classical Schur-Weyl construction. The same classification yields a cyclic Fock space of type B whose Gaussian operator has moments given by the Askey-Wimp-Kerov distribution through a new notion of cycles on pair-partitions. The method also reduces the corresponding classification problem for type D Coxeter groups to the type B case.

Core claim

The signed reflection function φ_{q+,q-} is positive definite if and only if it is an extreme character of the infinite hyperoctahedral group, and the corresponding set of parameters q+,q- is classified explicitly. The associated representations arise from the natural action of the finite hyperoctahedral group B(n) on the tensor product of n copies of a vector space, which supplies the two-parameter analog of the Schur-Weyl duality. Applications include a cyclic Fock space of type B and a Gaussian operator whose moments are identified with the Askey-Wimp-Kerov distribution by counting cycles on pair-partitions.

What carries the argument

The two-parameter signed reflection function φ_{q+,q-}, defined as a bivariate refinement of reflection length that records long and short reflections separately.

If this is right

  • The classification produces explicit representations of the infinite hyperoctahedral group via tensor-product actions of the finite groups B(n).
  • A cyclic Fock space of type B is obtained that generalizes the one-parameter construction previously known for type A.
  • A Gaussian operator on this Fock space has all moments determined by the Askey-Wimp-Kerov distribution through the enumeration of cycles on pair-partitions.
  • The same classification reduces the problem of extreme characters for type D Coxeter groups to the type B case.

Where Pith is reading between the lines

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

  • The separation of reflection lengths into two families may extend to other infinite Coxeter groups that possess more than one conjugacy class of reflections.
  • The cycle enumeration on pair-partitions supplies a combinatorial model that could be used to compute higher moments or joint distributions in noncommutative probability spaces of type B.
  • The tensor-product construction suggests a possible multi-parameter deformation of Schur-Weyl duality for other infinite wreath products.
  • One could test the boundary cases of the classified parameter region by direct computation of the positive-definiteness matrix for small n.

Load-bearing premise

The bivariate refinement of reflection length that separates long and short reflections can be used to test positive definiteness directly against the list of extreme characters.

What would settle it

Compute the matrix of inner products for φ_{q+,q-} at a pair (q+,q-) outside the classified set and check whether the resulting form is positive definite; the claim fails if any such pair yields a positive definite function that is not an extreme character.

Figures

Figures reproduced from arXiv: 2104.14530 by Maciej Do{\l}\k{e}ga, Marek Bo\.zejko, \'Swiatos{\l}aw R. Gal, Wiktor Ejsmont.

Figure 1
Figure 1. Figure 1: Coxeter diagram for B(n). One can also define the group B(n) as wreath product Z2 ≀ Sn with multiplication given by (g1, . . . , gn; σ) · (g ′ 1 , . . . , g′ n ; σ ′ ) = (g1g ′ σ−1(1), . . . , gng ′ σ−1(n) ; σσ′ ), where gi , g′ i ∈ {1, −1}, σ, σ′ ∈ Sn. We refer to this presentation as the signed model. In this model s0 corresponds to ((−1, 1, . . . , 1), id) and si corresponds to ((1, . . . , 1),(i i+ 1))… view at source ↗
Figure 2
Figure 2. Figure 2: Relation between q+, q− and M, N. In the left hand side we fix ǫ = 1, N = 4 and we look at the evolution of M and in the right hand side we fix ǫ = −1, M = 2 and we look at the evolution of N. Following [Poi98] we are going to relate the representation theory of B(n) with the theory of symmetric functions. We denote by x (y respectively) the infinite alphabet x1, x2 . . . (y1, y2 . . .). Let ǫ ∈ {+, −} and… view at source ↗
Figure 3
Figure 3. Figure 3: Two examples of connected pairs. The bottom non-cr [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An example of a partition π ∈ Psym 2 (10) with one negative and two positive cycles. Remark 4.11. 1). In the case of a positive cycle σ = (l1, r1)(r1, l1) it should be understood that the pairs (l1, r1),(r1, l1) lie in both partitions π and π. ˆ 2). Our definition of cycles in P sym 2 (n) is quite similar to the definition of partitions of type B [Rei97, Section 2], but in our situation a zero block (which… view at source ↗
Figure 5
Figure 5. Figure 5: The partition π has one semi-cycle σ1 of length 3 with the edges of the negative part l(σ − 1 ) = 8, r(σ − 1 ) = 1, one semi-cycle σ2 of length 2 with the edges of the negative part l(σ − 2 ) = 5, r(σ − 1 ) = 2, and one semi-cycle σ3 of length 1 with only one edge of the negative part r(σ − 3 ) = 7. 4.3. Gaussian operator. We construct generalized cyclic Gaussian operators of type B. They are given by the … view at source ↗
Figure 6
Figure 6. Figure 6: The visualization of the action of α0(xk ⊗ xk) [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The visualization of the action of γq− (xk) Case 2d). Suppose that γq− (xk ⊗ xk) acts on the tensor product (24), which gives us hxk, xl(σ + p ) ihxk , xl(σ − p ) i with coefficient q−. This situation is fully analogous to the Case 2c) and we proceed as shown in [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The visualization of the action of βq+ (xk) on (23) Case 3a). If βq+ (xk ⊗xk) acts on the tensor product (23) then there are 2(p−1) new terms which appear in (19). The action on the i th term in the tensor product (23), for 1 ≤ i ≤ p − 1 contributes to the (a) inner product hxk, xl(σ + i ) ihxk , xl(σ − i ) i with coefficient q+ if we apply the action of q+Ji ; (b) inner product hxk, xl(σ − i ) ihxk , xl(σ… view at source ↗
Figure 9
Figure 9. Figure 9: The visualization of the action of βq+ (xk) on (24) We emphasize that in all of the above steps r(σ ± p ) is the most right or left point representing vector in the tensor product (23) and (24). By using this point we create the partition πˆ˜. This explains why πˆ˜ is non-crossing and don’t cover singletons. Note that as π runs over P sym 1,2;ǫ (k−1), every set partition π˜ ∈ Psym 1,2;ǫ (k) appears exactly… view at source ↗
read the original abstract

We introduce a two-parameter function $\phi_{q_+,q_-}$ on the infinite hyperoctahedral group, which is a bivariate refinement of the reflection length keeping track of the long and the short reflections separately. We show that this signed reflection function $\phi_{q_+,q_-}$ is positive definite if and only if it is an extreme character of the infinite hyperoctahedral group and we classify the corresponding set of parameters $q_+,q_-$. We construct the corresponding representations through a natural action of the hyperoctahedral group $B(n)$ on the tensor product of $n$ copies of a vector space, which gives a two-parameter analog of the classical construction of Schur--Weyl. We apply our classification to construct a cyclic Fock space of type B generalizing the one-parameter construction in type A found previously by Bo\.zejko and Guta. We also construct a new Gaussian operator acting on the cyclic Fock space of type B and we relate its moments with the Askey--Wimp--Kerov distribution by using the notion of cycles on pair-partitions, which we introduce here. Finally, we explain how to solve the analogous problem for the Coxeter groups of type D by using our main result.

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 / 3 minor

Summary. The paper introduces a two-parameter function φ_{q+,q-} on the infinite hyperoctahedral group B_∞ as a bivariate refinement of the reflection length that separately tracks long and short reflections. It proves that φ_{q+,q-} is positive definite if and only if it is an extreme character of B_∞ and classifies the admissible parameters q+, q-. The corresponding representations are realized by a natural action of the finite groups B(n) on an n-fold tensor product of a two-parameter vector space, yielding a two-parameter analog of the classical Schur–Weyl construction. The classification is applied to construct a cyclic Fock space of type B (generalizing the one-parameter type-A construction of Bożejko–Guta), to define a new Gaussian operator on this space whose moments are identified with the Askey–Wimp–Kerov distribution via a new notion of cycles on pair-partitions, and to solve the analogous problem for Coxeter groups of type D.

Significance. If the classification and constructions hold, the work supplies a two-parameter extension of known results on extreme characters and positive-definite functions in the asymptotic representation theory of infinite Coxeter groups of type B/C. The Schur–Weyl-type tensor-product realization and the explicit link between the Gaussian operator moments and the Askey–Wimp–Kerov distribution (via the introduced cycles on pair-partitions) are concrete strengths that could be useful for further work on Fock spaces and non-commutative probability.

minor comments (3)
  1. [Introduction] The definition of the signed reflection function φ_{q+,q-} (Introduction) should include an explicit formula or generating-function expression before the positive-definiteness statement is invoked.
  2. [Applications] In the section on the Gaussian operator, the precise relation between the moments and the Askey–Wimp–Kerov distribution via cycles on pair-partitions should be stated as a numbered theorem or proposition rather than only in the text.
  3. Notation for the infinite group (B_∞ versus the inductive limit of B(n)) and for the two-parameter vector space should be fixed consistently throughout.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript and for the positive assessment of its significance. The recommendation of minor revision is noted. As the major comments section contains no specific points, we have no individual comments to address point-by-point and will make any appropriate minor adjustments in the revised version.

Circularity Check

0 steps flagged

Minor self-citation present but derivation remains self-contained

full rationale

The paper defines φ_{q+,q-} directly as a bivariate refinement of reflection length on the infinite hyperoctahedral group, constructs the associated representations explicitly via a two-parameter tensor-product action of B(n) that generalizes the classical Schur-Weyl construction, and derives the positive-definiteness condition together with the parameter classification from this representation-theoretic setup and the inductive-limit argument. The single reference to prior one-parameter work by Bożejko and Guta supplies only contextual generalization for the type-A case and does not supply any load-bearing step, uniqueness theorem, or fitted input for the type-B/C classification. No self-definitional reductions, fitted inputs renamed as predictions, or ansätze smuggled via citation appear in the derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the definition of the two-parameter function and standard representation theory axioms; the new elements are the classification and the combinatorial tool for the application.

free parameters (1)
  • q_+, q_-
    These are the parameters whose admissible values are classified by the main theorem rather than being fitted to external data.
axioms (1)
  • standard math A function on a group is positive definite if and only if it is the character of a unitary representation
    This is a standard fact from representation theory used to link positive definiteness to extreme characters.
invented entities (1)
  • cycles on pair-partitions no independent evidence
    purpose: To express the moments of the Gaussian operator in terms of the Askey-Wimp-Kerov distribution
    A new combinatorial concept introduced in the paper for this application.

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