Pith. sign in

REVIEW 2 major objections 3 minor 1 cited by

From Weingarten calculus for real Grassmannians to deformations of monotone Hurwitz numbers and Jucys-Murphy elements

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Real-Grassmannian Weingarten functions expand into weighted monotone factorisations, providing a combinatorial basis for b- and t-deformed monotone Hurwitz numbers and for b-deformed Jucys–Murphy operators.

desk verdict Strong new Weingarten/Hurwitz results in Sections 3–4; Section 5's Jucys–Murphy proof has a genuine algebraic error and needs repair before acceptance. read the letter →

arxiv 2506.04002 v1 pith:EZ3IEIYN submitted 2025-06-04 math.CO math-phmath.MPmath.RT

Xavier Coulter , Norman Do This is my paper
classification math.COmath-phmath.MPmath.RT MSC 05A1505E1015B5260B20
keywords WeingartencalculusrealGrassmanniansmonotoneHurwitznumberspairpartitionsJucys-MurphyelementsJacksymmetricfunctionsVirasoroconstraintslarge-Nexpansion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

At the core of the paper is a Weingarten calculus for the real Grassmannian, realised as the space of idempotent real symmetric matrices of rank M. The authors prove that every polynomial integral over this space expands in powers of $1/N$, and that in the large-$N$ regime with $M/N$ fixed, the coefficients are weighted counts of monotone factorisations of pair partitions, with weight $t$ raised to the hive number. These $t$-deformed counts are then absorbed into a two-parameter $bt$-monotone Hurwitz number, where the $b$-parameter records a flip number; the same construction, at $b=0$ and $b=1$, recovers the unitary and orthogonal Weingarten calculi. Along the way the paper introduces $b$-deformed Jucys–Murphy operators and conjectures that they diagonalise on a basis indexed by standard Young tableaux with eigenvalues given by $b$-contents, linking the deformation to Jack symmetric functions. A sympathetic reader should care because the paper gives a combinatorial bridge from matrix-integral Weingarten calculus to monotone Hurwitz numbers and to the representation-theoretic structures of the symmetric group.

What carries the argument

The load-bearing object is the weight function $\omega^{(b)}$ on pairs of pair partitions, defined by a charge assignment on the cycles of the graph $\Gamma(m)$ built from $m$ and the identity pair partition; it returns $0$, $1$, or $b$ and is independent of the transposition chosen. This weight function decorates the edges of the orthogonal Weingarten graph to produce the $b$-Weingarten graph, and adding the $t$-deformed type-$C$ edges produces the $bt$-Weingarten graph. The orthogonality relations for the corresponding Weingarten functions express each value as a walk count in these graphs; the large-$N$ expansion then becomes an enumeration of monotone factorisations of pair partitions, with the $b$ and $t$ parameters recording flip and hive numbers.

What would settle it

Compute $Wg_A(m)$ for two pair partitions $m, m'$ of the same coset-type at fixed finite $M$ and $N$, say $M=3$, $N=5$, $k=3$, by direct numerical integration over $A(M,N)$; if the two values differ, the coset-type invariance fails and the large-$N$ expansion theorem collapses. Alternatively, verify for small $N$ and $M$ that the coefficient of $N^{-1}$ in Theorem 3.11 matches an explicit enumeration of monotone factorisations of a given pair partition, with a mismatch falsifying the expansion.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 3.11: for every pair partition $m$ of size $k$, the Weingarten function $Wg_A(m)$ has the large-$N$ expansion $Wg_A(m) = \frac{1}{(1-t)^k} \sum_{r\ge 0} \vec{h}_r^{(t)}(m) \left(-\frac{1}{N}\right)^r$, where $\vec{h}_r^{(t)}(m) \in \mathbb{Z}[t]$ counts monotone factorisations of $m$ of length $r$ with weight $t^{\mathrm{hive}(\tau)}$. Together with Propositions 4.7 and 4.17, this identifies the $b$- and $bt$-Weingarten functions with $b$- and $bt$-monotone Hurwitz numbers in their large-$N$ expansions, and shows that the $bt$-monotone Hurwitz numbers count connected monotone factorisations weighted by $b^{\mathrm{flip}(\tau)} t^{\mathrm{hive}(\tau)}$. The same formalism yields a Virasoro-constrained partition function and a cut-join-flip recursion, and motivates a $b$-deformed family of Jucys–Murphy operators for which the paper proves one symmetric-function statement and formulates several conjectures connecting them to Jack functions.

Load-bearing premise

The Weingarten function $Wg_A(m)$ depends only on the coset-type of the pair partition $m$, a symmetry inherited from $A^T = A$ and $O(N)$-invariance; every orthogonality relation and large-$N$ expansion in the paper relies on this descent from pair partitions to partitions.

Editorial extensions

If this is right

  • Values of the Weingarten function $Wg_A(m)$ are explicit rational functions in $N$ and $M$, computable recursively from the base case, so polynomial integrals over the real Grassmannian are algorithmically evaluable.
  • The coefficients $\vec{h}_r^{(t)}(m)$ are polynomials in $t$ with integer coefficients, so the $t$-deformed orthogonal monotone Hurwitz numbers form a genuine $\mathbb{Z}[t]$-valued deformation of monotone Hurwitz numbers.
  • The $b$-Weingarten calculus gives a combinatorial interpretation of the previously defined $b$-monotone Hurwitz numbers as monotone factorisations weighted by flip number, with no matrix integral needed for general $b$.
  • The $bt$-monotone Hurwitz numbers satisfy Virasoro constraints and a cut-join-flip recursion, giving an effective computation method and a common generalisation of the classical cut-join recursion.
  • If the paper's real-rootedness and interlacing conjectures hold, every $bt$-monotone Hurwitz number is a real-rooted polynomial in $t$ whose coefficients are symmetric and unimodal, and it interlaces its one-part increments.
  • The $b$-deformed Jucys–Murphy operators, if their conjectures hold, recover a Gelfand–Tsetlin-type basis for the space $X^{(k)}$ with eigenvalues given by $b$-contents, giving a deformation of the classical representation-theoretic picture for the symmetric group.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The $b$-deformation is not tied to any matrix integral for general $b$, so the particular choice of weight function $\omega^{(b)}$ is one of many; it is plausible that other weight functions satisfying the same interpolation properties would produce different J-operators, and the paper's conjectures may depend on this choice.
  • If the conjectures on the $b$-deformed Jucys–Murphy operators hold, these operators provide a combinatorial model for the Gelfand–Tsetlin basis with $b$-contents, which could give a new route toward the open conjectures on Jack functions that the paper cites.
  • The point $b = -1/2$, where the construction subsumes the symplectic Weingarten calculus, suggests a testable specialisation of the $bt$-monotone Hurwitz numbers that should inherit real-rootedness if the general conjecture holds.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper develops a Weingarten calculus for the real Grassmannian, realized as the space of N×N idempotent real symmetric matrices of rank M. It defines a Weingarten function Wg_A, proves a convolution formula, orthogonality relations, and a large-N expansion whose coefficients are weighted enumerations of monotone factorisations of pair partitions, with weights recording a 'hive number' (Theorem A). It then introduces a purely combinatorial 'b-Weingarten calculus' interpolating between unitary (b=0) and orthogonal (b=1) cases, and combines the b- and t-deformations into a bt-Weingarten calculus. The bt-monotone Hurwitz numbers are shown to satisfy Virasoro constraints, a cut-join-flip recursion, and a combinatorial interpretation via monotone factorisations weighted by flip and hive numbers (Theorem B). The final section defines b-deformed Jucys–Murphy operators, states several conjectures about their representation-theoretic and Jack-function properties, and proves partial results (Proposition 5.9). The central Weingarten/Hurwitz part of the paper appears sound, but the proof of Proposition 5.9 contains a concrete algebraic error in the expansion of (M+J)/(N+J), and Proposition 5.6 is stated with a mismatch between the b- and bt-Weingarten functions.

Significance. If the main theorems stand, the paper gives a new matrix-integral realization of b-monotone Hurwitz numbers and introduces a two-parameter bt-deformation with explicit Virasoro constraints, a cut-join-flip recursion, and a clean combinatorial interpretation. The b-Weingarten graph is a genuinely new interpolating object, and the paper offers extensive computational evidence for the real-rootedness and interlacing conjectures, including over 4000 checks across many values of g, n, and b. The connection to Jack functions via b-deformed Jucys–Murphy operators is suggestive and potentially important, but the current proof of the key supporting result (Proposition 5.9) is invalid as written, so the Jucys–Murphy/Jack part requires repair before the significance of that section can be fully credited.

major comments (2)
  1. [Section 5.1, Proposition 5.6] Proposition 5.6 equates the b-Weingarten function Wg^(b)(m), which by Definition 4.5 is an element of R[b][[N^{-1}]] with no dependence on M, to a product of factors (M+J_i)/(N+J_i) that depends on M. The intended statement must involve the bt-Weingarten function Wg^(bt) of Definition 4.12, whose orthogonality relations are Proposition 4.13. Moreover, the proposition is stated without proof even though it is the starting point for the expansion used in the proof of Proposition 5.9; a corrected statement with a proof (by induction along the lines of Proposition 3.12) is needed.
  2. [Section 5.2, proof of Proposition 5.9, equation (23)] The expansion of (M+J)/(N+J) used in equation (23) is algebraically incorrect. With ℏ=-1/N and t=1-N/M, the exact identity is (M+J)/(N+J) = (1-t)^{-1}(1-(1-t)ℏJ)/(1-ℏJ) = (1-t)^{-1}(1+tℏJ/(1-ℏJ)), whereas the paper displays (1-t)^{-1}(1+tℏJ)/(1-ℏJ). These differ already in the coefficient of ℏJ: for t=0 the left side equals 1 while the displayed factor equals (1-ℏJ)^{-1}. Since equations (23)–(26) compare coefficients of ℏ^r t^ℓ to extract identities for the J-operators, the derivation of equation (22) and hence of equation (21) is invalid as written. The proof needs to be redone with the correct expansion; the final identities may still be true, but they are not established by this argument.
minor comments (3)
  1. [Definition 3.8] The definition says a monotone factorisation is a sequence of transpositions in S_k, but these transpositions act on pair partitions of {1,...,2k}; the group should presumably be S_{2k}.
  2. [Section 4.3, Proposition 4.20] Proposition 4.20 is stated as a formal result but its proof is omitted with only a reference to a previous proof in [14]; either include the proof or explicitly label the statement as following verbatim from [14, Proposition 3.9].
  3. [Throughout] There are several small typos, including 'on the the real Grassmannian' at the start of Section 3, 'positive change' instead of 'positive charge' in Definition 4.1, and the reference in Section 5.1 to 'Proposition 4.13' when the b-case orthogonality relations are in Proposition 4.6.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central Weingarten, b- and bt-Hurwitz identities are derived from their own recursions, not from their conclusions; only minor self-citation and an unproven or erroneous Proposition 5.6 are flagged.

full rationale

The paper's central chain is self-contained. Wg_A is defined by a genuine matrix integral (Definition 3.3), the convolution formula and orthogonality relations are proved by lifting to O(N) (Proposition 3.4 and Theorem 3.5), and the large-N expansion is derived from the Weingarten graph and then converted into a monotone-factorisation enumeration (Theorem 3.11), not assumed. The b- and bt-deformations are introduced combinatorially (Definitions 4.1, 4.3, 4.5, 4.11, and 4.12), and the equalities to b- and bt-monotone Hurwitz numbers (Propositions 4.7 and 4.17) are established by induction showing that both sides satisfy the same cut/join/flip recursion obtained from the Jack-function Virasoro constraints of [2] and [7]. No parameter is fitted to the target numbers, and the type-C edge weight of 1 is fixed by the path-generating-function identity (17) rather than by importing a target value. The only self-citation that is load-bearing for a secondary result is Proposition 4.20, whose proof is omitted and deferred to the authors' prior work [14]; this is minor and does not affect the central claims. Per the review rule, two non-circular issues are flagged explicitly: Proposition 5.6 is stated 'without proof' and is then used in the proof of Proposition 5.9, and the expansion of (M+J)/(N+J) as (1+ℏtJ)/(1−ℏJ) near equations (23)–(26) is algebraically incorrect, so the proof of Proposition 5.9 requires repair. These are correctness risks, not instances of circular derivation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The paper introduces no fitted parameters. The main axioms are standard theorems from orthogonal Weingarten calculus and the b-monotone Hurwitz literature, plus the domain assumption about Haar measure on the Grassmannian. The invented entities are formal constructions whose consistency is proven in the paper; none are empirical postulates.

assumptions (5)
  • standard math The orthogonal Weingarten calculus (convolution formula and orthogonality relations, Theorems 2.8-2.9) is valid.
    Invoked in the proof of Proposition 3.4 and throughout Section 3.
  • domain assumption The space A(M,N) is a compact homogeneous space with a normalized O(N)-invariant Haar measure.
    Definition 3.1 and the definition of integrals over A(M,N) rely on this.
  • standard math The cited result [7, Theorem 2.7] on PDEs for b-deformed weighted Hurwitz numbers applies to Z^(bt).
    Used as a black box in the proof of Theorem 4.15.
  • standard math The Virasoro constraints for b-monotone Hurwitz numbers (from [2, Theorem 2.7]) are valid.
    Used in the proof of Proposition 4.7.
  • domain assumption The technical assumption k <= N does not affect the large-N expansions.
    Stated in Section 3 before Definition 3.3; the paper argues it is harmless.
invented entities (3)
  • b-Weingarten calculus independent evidence
    purpose: Interpolate between unitary and orthogonal Weingarten calculus for general parameter b, without reference to matrix integrals.
    It recovers the unitary and orthogonal cases at b=0 and b=1, and its large-N coefficients are identified with the known b-monotone Hurwitz numbers.
  • b-deformed Jucys-Murphy operators J_i independent evidence
    purpose: Generalize Jucys-Murphy elements to act on pair partitions, conjecturally diagonalized by b-contents.
    The conjectures give explicit eigenvalues and bases that can be checked for larger k; verified for k up to 5.
  • weight function omega_b
    purpose: Assign weights {0,1,b} to edges of the Weingarten graph to define the b-deformation.
    This is an ad hoc construction; the paper notes there are many possible choices. Its properties are internally motivated but no external falsifiable test is given beyond the derived results.

how reviews work

0 comments
Cite this review

Pith. "Pith review of From Weingarten calculus for real Grassmannians to deformations of monotone Hurwitz numbers and Jucys-Murphy elements." pith.science (2026). https://pith.science/paper/EZ3IEIYN

@misc{pith2026250604002,
  author       = {Pith},
  title        = {Pith review of: From Weingarten calculus for real Grassmannians to deformations of monotone Hurwitz numbers and Jucys-Murphy elements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EZ3IEIYN}},
  note         = {Machine review of arXiv:2506.04002}
}
abstract

The present work is inspired by three interrelated themes: Weingarten calculus for integration over unitary groups, monotone Hurwitz numbers which enumerate certain factorisations of permutations into transpositions, and Jucys-Murphy elements in the symmetric group algebra. The authors and Moskovsky recently extended this picture to integration on complex Grassmannians, leading to a deformation of the monotone Hurwitz numbers to polynomials that are conjectured to satisfy remarkable interlacing phenomena. In this paper, we consider integration on the real Grassmannian $\mathrm{Gr}_\mathbb{R}(M,N)$, interpreted as the space of $N \times N$ idempotent real symmetric matrices of rank $M$. We show that in the regime of large $N$ and fixed $\frac{M}{N}$, such integrals have expansions whose coefficients are variants of monotone Hurwitz numbers that are polynomials in the parameter $t = 1 - \frac{N}{M}$. We define a "$b$-Weingarten calculus", without reference to underlying matrix integrals, that recovers the unitary case at $b = 0$ and the orthogonal case at $b = 1$. The $b$-monotone Hurwitz numbers, previously introduced by Bonzom, Chapuy and Dolega, arise naturally in this context as monotone factorisations of pair partitions. The $b$- and $t$-deformations can be combined to form a common generalisation, leading to the notion of $bt$-monotone Hurwitz numbers, for which we state several results and conjectures. Finally, we introduce certain linear operators inspired by the aforementioned $b$-Weingarten calculus that can be considered as $b$-deformations of the Jucys-Murphy elements in the symmetric group algebra. We make several conjectures regarding these operators that generalise known properties of the Jucys-Murphy elements and make a connection to the family of Jack symmetric functions.

Figures

Figures reproduced from arXiv: 2506.04002 by the authors.

Figure 1
Figure 1. The orthogonal Weingarten graph G O, restricted to P0 ⊔ P1 ⊔ P2 ⊔ P3. Each solid edge represents two type A edges, one in each direction. Each dashed edge represents one type B edge, directed down the page. Each blue vertex should also have a directed loop, although these are not depicted to avoid cluttering the diagram. By virtue of the construction of the orthogonal Weingarten graph from the orthogonality relation… view at source ↗
Figure 2
Figure 2. The b-contents (left) and the b-hook lengths (right) of the partition (5, 4, 4, 2). Note that at b = 0, one recovers the usual notions of content, hook length, and hook product, which appears in the hook length formula for the dimension of the Specht module labelled by λ. dim λ = |λ|! hook0(λ) 11 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. The graph Γ(m) corresponding to the pair partition m = (1 8 | 2 6 | 3 9 | 4 10 | 5 7) ∈ P5. The edges corresponding to m are drawn in blue, while the edges corresponding to e5 are drawn in red. Each vertex is labelled by its charge. We use the function ω (b) to assign weights to the directed edges in the orthogonal Weingarten graph to obtain the b-Weingarten graph as follows. Definition 4.3. The b-Weingarten graph G… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The b-Weingarten graph G (b) , restricted to P0 ⊔ P1 ⊔ P2 ⊔ P3. Each solid edge represents two type A edges, one in each direction. Each dashed edge represents one type B edge, directed down the page. An arrowhead on an edge indicates that the weight b is assigned to t…

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A refined twist on Hurwitz numbers

    math.CO 2025-08 conditional novelty 8.0 of 10

    A two-parameter CJT refinement of Jucys-Murphy theory interpolates Schur and zonal actions, yields cut-and-join recursions and tropicalizations of b-Hurwitz numbers, and proves piecewise polynomiality of (1+b) times t...

Reference graph

Works this paper leans on

35 extracted references · 33 canonical work pages · cited by 1 Pith paper

  1. [2]

    b-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, andO(N )-BGW integral

    Valentin Bonzom, Guillaume Chapuy, and Maciej Dołęga. b-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, andO(N )-BGW integral. Int. Math. Res. Not. IMRN, (14):12172–12230, 2023

    work page 2023

  2. [7]

    Chidambaram, Maciej Dołęga, and Kento Osuga

    Nitin K. Chidambaram, Maciej Dołęga, and Kento Osuga. b-Hurwitz numbers from Whittaker vectors for W-algebras. arXiv:2401.12814 [math.AG], 2024

    arXiv 2024

  3. [1]

    Some conjectures on the Schur expansion of Jack polynomials

    Per Alexandersson, James Haglund, and George Wang. Some conjectures on the Schur expansion of Jack polynomials. J. Comb., 12(2):215–233, 2021

    work page 2021

  4. [3]

    Relating ordinary and fully simple maps via monotone Hurwitz numbers

    Gaëtan Borot, Séverin Charbonnier, Norman Do, and Elba Garcia-Failde. Relating ordinary and fully simple maps via monotone Hurwitz numbers. Electron. J. Combin., 26(3):Paper No. 3.43, 24, 2019

    work page 2019

  5. [4]

    Functional relations for higher-order free cumulants

    Gaëtan Borot, Séverin Charbonnier, Elba Garcia-Failde, Felix Leid, and Sergey Shadrin. Functional relations for higher-order free cumulants. arXiv:2112.12184 [math.OA], 2021

    work page Pith review arXiv 2021

  6. [5]

    Unimodal polynomials arising from symmetric functions

    Francesco Brenti. Unimodal polynomials arising from symmetric functions. Proc. Amer. Math. Soc. , 108(4):1133–1141, 1990

    work page 1990

  7. [6]

    Hermitian matrix model free energy: Feynman graph technique for all genera

    Leonid Chekhov and Bertrand Eynard. Hermitian matrix model free energy: Feynman graph technique for all genera. J. High Energy Phys., (3):014, 18, 2006

    work page 2006

  8. [8]

    b-Hurwitz numbers from refined topological recursion

    Nitin Kumar Chidambaram, Maciej Dołęga, and Kento Osuga. b-Hurwitz numbers from refined topological recursion. arXiv:2412.17502 [math.CO], 2024

    arXiv 2024

Show all 35 references
  1. [9]

    Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability

    Benoît Collins. Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability. Int. Math. Res. Not., (17):953–982, 2003

    2003

  2. [10]

    On some properties of orthogonal Weingarten functions

    Benoît Collins and Sho Matsumoto. On some properties of orthogonal Weingarten functions. J. Math. Phys., 50(11):113516, 14, 2009

    2009

  3. [11]

    Weingarten calculus via orthogonality relations: new applications

    Benoît Collins and Sho Matsumoto. Weingarten calculus via orthogonality relations: new applications. ALEA Lat. Am. J. Probab. Math. Stat., 14(1):631–656, 2017

    2017

  4. [12]

    The Weingarten calculus

    Benoît Collins, Sho Matsumoto, and Jonathan Novak. The Weingarten calculus. Notices Amer. Math. Soc., 69(5):734–745, 2022

    2022

  5. [13]

    Integration with respect to the Haar measure on unitary, orthogonal and symplectic group

    Benoît Collins and Piotr Śniady. Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Comm. Math. Phys., 264(3):773–795, 2006

    2006

  6. [14]

    Integration on complex Grassmannians, deformed monotone Hurwitz numbers, and interlacing phenomena

    Xavier Coulter, Norman Do, and Ellena Moskovsky. Integration on complex Grassmannians, deformed monotone Hurwitz numbers, and interlacing phenomena. arXiv:2308.04015 [math.CO], 2023

    2023 arXiv

  7. [15]

    Norman Do, Alastair Dyer, and Daniel V. Mathews. Topological recursion and a quantum curve for monotone Hurwitz numbers. J. Geom. Phys., 120:19–36, 2017

    2017

  8. [16]

    Seelinger

    Stephen Doty, Aaron Lauve, and George H. Seelinger. Canonical idempotents of multiplicity-free families of algebras. Enseign. Math., 64(1-2):23–63, 2018

    2018

  9. [17]

    Eynard and N

    B. Eynard and N. Orantin. Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys., 1(2):347–452, 2007. 39

    2007

  10. [18]

    I. P. Goulden, Mathieu Guay-Paquet, and Jonathan Novak. Monotone Hurwitz numbers in genus zero. Canad. J. Math., 65(5):1020–1042, 2013

    2013

  11. [19]

    I. P. Goulden, Mathieu Guay-Paquet, and Jonathan Novak. Monotone Hurwitz numbers and the HCIZ integral. Ann. Math. Blaise Pascal, 21(1):71–89, 2014

    2014

  12. [20]

    I. P. Goulden and D. M. Jackson. Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions. Trans. Amer. Math. Soc., 348(3):873–892, 1996

    1996

  13. [21]

    Mathieu Guay-Paquet and J. Harnad. 2D Toda τ-functions as combinatorial generating functions. Lett. Math. Phys., 105(6):827–852, 2015

    2015

  14. [22]

    Jack symmetric functions and some combinatorial properties of Young symmetrizers

    Phil Hanlon. Jack symmetric functions and some combinatorial properties of Young symmetrizers. J. Combin. Theory Ser. A, 47(1):37–70, 1988

    1988

  15. [23]

    A class of symmetric polynomials with a parameter.Proc

    Henry Jack. A class of symmetric polynomials with a parameter.Proc. Roy. Soc. Edinburgh Sect. A, 69:1–18, 1970/71

    1970

  16. [24]

    A.-A. A. Jucys. Symmetric polynomials and the center of the symmetric group ring. Rep. Mathematical Phys., 5(1):107–112, 1974

    1974

  17. [25]

    Jack polynomials and free cumulants

    Michel Lassalle. Jack polynomials and free cumulants. Adv. Math., 222(6):2227–2269, 2009

    2009

  18. [26]

    I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford Classic T exts in the Physical Sciences. The Clarendon Press, Oxford University Press, New York, second edition, 2015. With contribution by A. V. Zelevinsky and a foreword by Richard Stanley, Reprint of the 2008...

    2015

  19. [27]

    Jucys-Murphy elements, orthogonal matrix integrals, and Jack measures

    Sho Matsumoto. Jucys-Murphy elements, orthogonal matrix integrals, and Jack measures. Ramanujan J., 26(1):69–107, 2011

    2011

  20. [28]

    Weingarten calculus for matrix ensembles associated with compact symmetric spaces

    Sho Matsumoto. Weingarten calculus for matrix ensembles associated with compact symmetric spaces. Random Matrices Theory Appl., 2(2):1350001, 26, 2013

    2013

  21. [29]

    Jucys-Murphy elements and unitary matrix integrals

    Sho Matsumoto and Jonathan Novak. Jucys-Murphy elements and unitary matrix integrals. Int. Math. Res. Not. IMRN, (2):362–397, 2013

    2013

  22. [30]

    G. E. Murphy. A new construction of Young’s seminormal representation of the symmetric groups. J. Algebra, 69(2):287–297, 1981

    1981

  23. [31]

    Jonathan I. Novak. Jucys-Murphy elements and the unitary Weingarten function. In Noncommutative harmonic analysis with applications to probability II, volume 89 ofBanach Center Publ., pages 231–235. Polish Acad. Sci. Inst. Math., Warsaw, 2010

    2010

  24. [32]

    A new approach to representation theory of symmetric groups

    Andrei Okounkov and Anatoly Vershik. A new approach to representation theory of symmetric groups. Selecta Math. (N.S.), 2(4):581–605, 1996

    1996

  25. [33]

    Refined topological recursion revisited: properties and conjectures

    Kento Osuga. Refined topological recursion revisited: properties and conjectures. Comm. Math. Phys., 405(12):Paper No. 296, 47, 2024

    2024

  26. [34]

    Richard P. Stanley. Some combinatorial properties of Jack symmetric functions.Adv. Math., 77(1):76–115, 1989

    1989

  27. [35]

    Asymptotic behavior of group integrals in the limit of infinite rank

    Don Weingarten. Asymptotic behavior of group integrals in the limit of infinite rank. J. Mathematical Phys., 19(5):999–1001, 1978. 40

    1978

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.