The Symplectic Schur Process

Cesar Cuenca; Matteo Mucciconi

arxiv: 2407.02415 · v3 · submitted 2024-07-02 · 🧮 math-ph · math.CO· math.MP· math.PR

The Symplectic Schur Process

Cesar Cuenca , Matteo Mucciconi This is my paper

Pith reviewed 2026-05-23 23:01 UTC · model grok-4.3

classification 🧮 math-ph math.COmath.MPmath.PR

keywords symplectic Schur processdeterminantal point processcorrelation kernelDown-Up Schur functionsBerele insertion algorithmlimit shapesymmetric Pearcey kernelpartitions

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The pith

The symplectic Schur process defines a determinantal point process on tuples of partitions with an explicit correlation kernel.

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

The paper defines the symplectic Schur process as a measure on tuples of partitions analogous to the Schur process but adapted to symplectic symmetry. It introduces Down-Up Schur functions and establishes new Cauchy-Littlewood-type identities for them. These identities allow the proof that the corresponding point process is determinantal with an explicit correlation kernel. The authors also study preserving dynamics and the asymptotics of a sampling algorithm based on Berele insertion, which produces a new limit kernel similar to the symmetric Pearcey kernel.

Core claim

We define a measure on tuples of partitions, called the symplectic Schur process, that should be regarded as the right analogue of the Schur process for the Cartan type C. The weights include universal symplectic characters and a novel family of Down-Up Schur functions for which we prove new Cauchy-Littlewood-type identities. Our main result is that the point process is determinantal with an explicit correlation kernel. We also present dynamics preserving the family and explore sampling via the Berele insertion algorithm, with asymptotics yielding a new kernel resembling the symmetric Pearcey kernel.

What carries the argument

The Down-Up Schur functions, defined to satisfy Cauchy-Littlewood-type identities that establish the determinantal property of the symplectic Schur process.

If this is right

The correlation kernel gives explicit formulas for all finite-dimensional marginals of the process.
Markov dynamics exist that leave the symplectic Schur processes invariant.
In a special case the process can be sampled using the Berele insertion algorithm.
Asymptotic analysis of the insertion process provides explicit formulas for the limit shape and bulk and edge fluctuations, including a new kernel in one regime.

Where Pith is reading between the lines

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

Similar constructions using appropriate characters could define processes for other Lie types.
The new kernel may connect to other symmetric random matrix models.
Numerical simulations of the insertion algorithm could verify the predicted limit shapes.
The framework might apply to studying fluctuations in other partition models with group symmetries.

Load-bearing premise

The Down-Up Schur functions satisfy the Cauchy-Littlewood-type identities needed to close the determinantal property.

What would settle it

A calculation of the two-point correlation function for the measure on a small instance of partitions that does not agree with the formula from the proposed kernel.

Figures

Figures reproduced from arXiv: 2407.02415 by Cesar Cuenca, Matteo Mucciconi.

**Figure 2.** Figure 2: Propagation of jump instructions to particles at level 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: The random update of a half-triangular array [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: Representation of the support of the symplectic Schur process. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: A depiction of the point process L(λ⃗) constructed through Berele insertion. In Section 6, we study a dynamics that preserves the symplectic Schur processes (1.10) and analyze its asymptotics. This dynamics can be alternatively sampled by a process on integer partitions λ (1) , λ(2) , λ(3) , . . . that adds or removes boxes from the partitions at each step, and that is equivalent to the particle dynamic… view at source ↗

**Figure 6.** Figure 6: (a) A sample of the point process L(λ⃗) = {(j, λ(j) i − i) ∶ 1 ≤ i ≤ n, 1 ≤ j ≤ k} obtained through the Berele insertion of k random words with law (8.2) in the alphabet An. Here we chose n = 200, k = 300. Red and orange curves denote respectively the upper and lower branch of the limit shape, computed in (8.10). The verical green line has abscissa n; the process to its left has the law of the oscillatin… view at source ↗

**Figure 7.** Figure 7: Plots of h ′ x,y(z) for real z and different values of x, y. From top-left to bottom-right these topologies correspond to the cases enumerated after Equation (8.6). The functions y±(x) above are the unique real numbers such that there exists some z0 ∈ R with h ′ x,y(x) (z0) = h ′′ x,y(x) (z0) = 0. (8.7) We label y+(x) the larger and y−(x) the smaller of the two, so that y+(x) ≥ y−(x). Moreover, we label th… view at source ↗

**Figure 8.** Figure 8: The contourplot of ζ → R {h 9 8 ,−1 (ζ)} − h 9 8 ,−1 (−1). Overlayed black and red curves are z and w integration curves deformed to be respectively of steepest descent and steepest ascent for R {h 9 8 ,−1 (ζ)}. The differences are the integration contours for the ξ variable and the cross terms, the one of our kernel being 2η η 2−ξ 2 . One can draw a parallel with the case of the Airy kernel, where the pre… view at source ↗

read the original abstract

We define a measure on tuples of partitions, called the symplectic Schur process, that should be regarded as the right analogue of the Schur process of Okounkov-Reshetikhin for the Cartan type C. The weights of our measure include factors that are universal symplectic characters, as well as a novel family of "Down-Up Schur functions" that we define and for which we prove new identities of Cauchy-Littlewood-type. Our main structural result is that the point process corresponding to the symplectic Schur process is determinantal and we find an explicit correlation kernel. We also present dynamics that preserve the family of symplectic Schur processes and explore an alternative sampling scheme, based on the Berele insertion algorithm, in a special case. Finally, we study the asymptotics of the Berele insertion process and find explicit formulas for the limit shape and fluctuations near the bulk and the edge. One of the limit regimes leads to a new kernel that resembles the symmetric Pearcey kernel.

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.

Desk Editor's Note private letter to a colleague

This paper defines the symplectic Schur process for type C, introduces Down-Up Schur functions with the needed identities, and derives an explicit determinantal kernel plus a new edge kernel.

read the letter

The central contribution is the construction of the symplectic Schur process as the Cartan type C version of the Okounkov-Reshetikhin Schur process. The authors introduce Down-Up Schur functions, prove Cauchy-Littlewood-type identities for them, and use those to establish that the point process is determinantal with an explicit correlation kernel. They also give dynamics that preserve the family and study the Berele insertion case, obtaining a limit shape plus bulk and edge fluctuations, one of which yields a new kernel resembling the symmetric Pearcey kernel. These pieces are presented as original and not reducible to prior work on classical Schur processes. The explicit formulas and the organization of the missing type C case are the parts that stand out as useful. The argument structure holds together once the new identities are granted; the stress-test found no internal inconsistency or circularity in how the identities close the determinantal property. The main soft spot is that the derivations of the identities and the kernel formula are technical, so a referee would need to check the steps carefully for gaps in the error terms or the precise range of the identities. Nothing in the abstract or the described structure suggests a load-bearing flaw. This is for people already working on random partitions, integrable probability, or generalizations across classical groups. A reader who needs the type C kernel or wants to extend the framework further will find concrete objects to work with. It is coherent on its own terms and deserves a serious referee. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript defines the symplectic Schur process, a probability measure on tuples of partitions whose weights involve universal symplectic characters together with a new family of Down-Up Schur functions. It proves Cauchy-Littlewood-type identities for the latter, establishes that the associated point process is determinantal, and supplies an explicit correlation kernel. The paper further constructs dynamics that preserve the family, examines Berele insertion as a sampling method in a special case, and derives the limit shape together with bulk and edge fluctuations for the insertion process, including a new kernel resembling the symmetric Pearcey kernel.

Significance. If the stated identities and kernel derivations hold, the work supplies the first explicit determinantal structure for a natural type-C analogue of the Okounkov-Reshetikhin Schur process. The explicit kernel, the new Down-Up Schur functions, and the asymptotic formulas (limit shape, fluctuations, and the Pearcey-like kernel) constitute concrete, usable results for integrable probability and random partitions in Cartan type C.

minor comments (3)

[§2] §2 (definition of Down-Up Schur functions): the generating-function definition is given, but the precise range of the auxiliary variables for which the functions are polynomials (as opposed to formal series) should be stated explicitly before the identities are invoked.
[§4] §4 (kernel derivation): the passage from the Cauchy-Littlewood identities to the explicit double-contour integral formula for the kernel would benefit from a one-paragraph summary of the residue computations that produce the final expression.
[§6] §6 (asymptotics): the statement of the limit-shape result should include the precise scaling regime (e.g., the relation between n and the parameters of the measure) under which the convergence holds.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring point-by-point response or manuscript changes at this stage.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper defines new objects (symplectic Schur process and Down-Up Schur functions), states and proves the required Cauchy-Littlewood-type identities for those objects, and derives the determinantal kernel from the identities. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation, or definitional tautology; the central claim follows from the newly established identities rather than presupposing them. This is a standard self-contained construction in combinatorial probability.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The construction rests on standard facts about partitions and characters plus the new Down-Up Schur functions whose identities are proved in the paper; no free parameters or invented physical entities are mentioned.

axioms (2)

standard math Standard properties of partitions and symmetric functions hold (implicit throughout).
Background for any work on Schur functions and their generalizations.
domain assumption Universal symplectic characters are well-defined and satisfy the required orthogonality or branching rules.
Invoked when defining the weights of the measure.

invented entities (1)

Down-Up Schur functions no independent evidence
purpose: To supply the novel weight factors that close the Cauchy-Littlewood identities for the symplectic measure.
New family introduced and studied in the paper; independent evidence would be external applications or further identities.

pith-pipeline@v0.9.0 · 5697 in / 1460 out tokens · 19645 ms · 2026-05-23T23:01:25.464091+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We define a measure on tuples of partitions, called the symplectic Schur process... novel family of 'Down-Up Schur functions'... prove new identities of Cauchy-Littlewood-type. Our main structural result is that the point process... is determinantal and we find an explicit correlation kernel.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The weights of our measure include factors that are universal symplectic characters, as well as a novel family of 'Down-Up Schur functions'

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Colored line ensembles for stochastic vertex models

[AB-2024] Amol Aggarwal and Alexei Borodin. Colored line ensembles for stochastic vertex models. Preprint; arXiv:2402.06868 (2024). [ASW-1952] Michael Aissen, Isaac Jacob Schoenberg and Anne M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Math´ ematique 2, no. 1 (1952), pp. 93–103. [AFHSA-2024] Seamus P. Albion, ...

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[BG-2016] Alexei Borodin and Vadim Gorin. Lectures on integrable probability. Probability and Statistical Physics in St. Petersburg 91 (2016), pp. 155–214. [BK-2010] Alexei Borodin and Jeffrey Kuan. Random surface growth with a wall and Plancherel measures for O(∞). Communications in Pure and Applied Mathematics 63 (2010), pp. 831–

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Random partitions and the gamma kernel

[BO-2005] Alexei Borodin and Grigori Olshanski. Random partitions and the gamma kernel. Advances in Mathematics 194, no. 1 (2005), pp. 141–202. [BO-2017] Alexei Borodin and Grigori Olshanski. Representations of the infinite symmetric group. Vol

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Integrable probability: From representation the- ory to Macdonald processes

[BP-2014] Alexei Borodin and Leonid Petrov. Integrable probability: From representation the- ory to Macdonald processes. Probability Surveys 11 (2014), pp. 1–58. [BP-2016] Alexei Borodin and Leonid Petrov. Nearest neighbor Markov dynamics on Macdonald processes. Advances in Mathematics 300 (2016), pp. 71–155. [BR-2005] Alexei Borodin and Eric M. Rains. Ey...

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Forrester, Taro Nagao and Graeme Honner

[FNH-1999] Peter J. Forrester, Taro Nagao and Graeme Honner. Correlations for the orthogonal- unitary and symplectic-unitary transitions at the hard and soft edges. Nuclear Physics B 553, no. 3 (1999), pp. 601–643. [GOY-2021] Shiliang Gao, Gidon Orelowitz and Alexander Yong. Newell-Littlewood numbers. Transactions of the American Mathematical Society 374,...

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Combinatorial Howe duality of symplectic type

[HK-2022] Taehyeok Heo and Jae-Hoon Kwon. Combinatorial Howe duality of symplectic type. Journal of Algebra, vol. 600 (2022), pp. 1–44. [Howe-1989] Roger Howe. Remarks on classical invariant theory. Transactions of the American Mathematical Society 313, no. 2 (1989), pp. 539–570. [IMS-2023] Takashi Imamura, Matteo Mucciconi and Tomohiro Sasamoto. Skew RSK...

work page 2022
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Discrete polynuclear growth and determinantal processes

[Joh-2003] Kurt Johansson. Discrete polynuclear growth and determinantal processes. Commu- nications in Mathematical Physics 242 (2003), pp. 277–329. [Joh-2005] Kurt Johansson. The arctic circle boundary and the Airy process. The Annals of Probability, vol. 33, no. 1 (2005), pp. 1–30. 54 [JN-2006] Kurt Johansson and Eric Nordenstam. Eigenvalues of GUE Min...

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On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters

[Koi-1989] Kazuhiko Koike. On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters. Advances in Mathematics 74, no. 1 (1989), pp. 57–86. [KT-1987] Kazuhiko Koike and Itaru Terada. Young-diagrammatic methods for the representa- tion theory of the classical groups of type Bn, Cn, Dn. Journ...

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Littlewood

[Lit-1958] Dudley E. Littlewood. Products and plethysms of characters with orthogonal, sym- plectic and symmetric groups. Canadian Journal of Mathematics 10 (1958), pp. 17–32. [Mac-1994] A. Murilo Santos Macˆ edo. Universal parametric correlations at the soft edge of the spectrum of random matrix ensembles. Europhysics Letters 26, no. 9 (1994),

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[Mac-1995] Ian G. Macdonald. Symmetric functions and Hall polynomials, 2nd edition. Oxford Mathematical Monographs, Oxford University Press, Oxford,

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Spin q-Whittaker polynomials and de- formed quantum Toda

[MP-2022] Matteo Mucciconi and Leonid Petrov. Spin q-Whittaker polynomials and de- formed quantum Toda. Communications in Mathematical Physics. Volume 389 (2022), pp. 1331–1416 [NOS-2024] Anton Nazarov, Olga Postnova and Travis Scrimshaw. Skew Howe duality and limit shapes of Young diagrams. Journal of the London Mathematical Society 109, no. 1 (2024), e1...

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Infinite wedge and random partitions

[Ok-2001b] Andrei Okounkov. Infinite wedge and random partitions. Selecta Mathematica 7, no. 1 (2001), pp. 57–81. [OP-2006] Andrei Okounkov and Rahul Pandharipande. Gromov-Witten theory, Hurwitz theory, and completed cycles. Annals of Mathematics 163, no. 2 (2006), pp. 517–560. [OR-2003] Andrei Okounkov and Nikolai Reshetikhin. Correlation function of Sch...

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[OV-1996] Grigori Olshanski and Anatoly Vershik

Birkh¨ auser Boston (2006). [OV-1996] Grigori Olshanski and Anatoly Vershik. Ergodic unitarily invariant measures on the space of infinite Hermitian matrices. American Mathematical Society Translation Series 2, no. 175 (1996), pp. 137–176. [PS-2002] Michael Pr¨ ahofer and Herbert Spohn. Scale Invariance of the PNG droplet and the Airy process. Journal of ...

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The Cauchy identity for Sp(2n)

[Sun-1990] Sheila Sundaram. The Cauchy identity for Sp(2n). Journal of Combinatorial Theory, Series A 53, no. 2 (1990), pp. 209–238. [Tin-2007] Peter Tingley. Three combinatorial models for ̂sln crystals, with applications to cylin- dric plane partitions. International Mathematics Research Notices, vol. 2007, no. 9, rnm143. [TW-2006] Craig Tracy and Harol...

work page 1990
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Refined Cauchy/Littlewood identities and six-vertex model partition functions: III

[WZJ-2016] Michael Wheeler and Paul Zinn-Justin. Refined Cauchy/Littlewood identities and six-vertex model partition functions: III. Deformed bosons. Advances in Mathematics, Vol. 299 (2016), pp. 543–600. [Whi-1952] Anne M. Whitney. A reduction theorem for totally positive matrices. Journal d’Analyse Math´ ematique 2, no. 1 (1952), pp. 88–92. [Zho-2022] C...

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[1] [1]

Colored line ensembles for stochastic vertex models

[AB-2024] Amol Aggarwal and Alexei Borodin. Colored line ensembles for stochastic vertex models. Preprint; arXiv:2402.06868 (2024). [ASW-1952] Michael Aissen, Isaac Jacob Schoenberg and Anne M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Math´ ematique 2, no. 1 (1952), pp. 93–103. [AFHSA-2024] Seamus P. Albion, ...

work page arXiv 2024

[2] [2]

Lectures on integrable probability

[BG-2016] Alexei Borodin and Vadim Gorin. Lectures on integrable probability. Probability and Statistical Physics in St. Petersburg 91 (2016), pp. 155–214. [BK-2010] Alexei Borodin and Jeffrey Kuan. Random surface growth with a wall and Plancherel measures for O(∞). Communications in Pure and Applied Mathematics 63 (2010), pp. 831–

work page 2016

[3] [3]

Random partitions and the gamma kernel

[BO-2005] Alexei Borodin and Grigori Olshanski. Random partitions and the gamma kernel. Advances in Mathematics 194, no. 1 (2005), pp. 141–202. [BO-2017] Alexei Borodin and Grigori Olshanski. Representations of the infinite symmetric group. Vol

work page 2005

[4] [4]

Integrable probability: From representation the- ory to Macdonald processes

[BP-2014] Alexei Borodin and Leonid Petrov. Integrable probability: From representation the- ory to Macdonald processes. Probability Surveys 11 (2014), pp. 1–58. [BP-2016] Alexei Borodin and Leonid Petrov. Nearest neighbor Markov dynamics on Macdonald processes. Advances in Mathematics 300 (2016), pp. 71–155. [BR-2005] Alexei Borodin and Eric M. Rains. Ey...

work page arXiv 2014

[5] [5]

Forrester, Taro Nagao and Graeme Honner

[FNH-1999] Peter J. Forrester, Taro Nagao and Graeme Honner. Correlations for the orthogonal- unitary and symplectic-unitary transitions at the hard and soft edges. Nuclear Physics B 553, no. 3 (1999), pp. 601–643. [GOY-2021] Shiliang Gao, Gidon Orelowitz and Alexander Yong. Newell-Littlewood numbers. Transactions of the American Mathematical Society 374,...

work page 1999

[6] [6]

Combinatorial Howe duality of symplectic type

[HK-2022] Taehyeok Heo and Jae-Hoon Kwon. Combinatorial Howe duality of symplectic type. Journal of Algebra, vol. 600 (2022), pp. 1–44. [Howe-1989] Roger Howe. Remarks on classical invariant theory. Transactions of the American Mathematical Society 313, no. 2 (1989), pp. 539–570. [IMS-2023] Takashi Imamura, Matteo Mucciconi and Tomohiro Sasamoto. Skew RSK...

work page 2022

[7] [7]

Discrete polynuclear growth and determinantal processes

[Joh-2003] Kurt Johansson. Discrete polynuclear growth and determinantal processes. Commu- nications in Mathematical Physics 242 (2003), pp. 277–329. [Joh-2005] Kurt Johansson. The arctic circle boundary and the Airy process. The Annals of Probability, vol. 33, no. 1 (2005), pp. 1–30. 54 [JN-2006] Kurt Johansson and Eric Nordenstam. Eigenvalues of GUE Min...

work page 2003

[8] [8]

On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters

[Koi-1989] Kazuhiko Koike. On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters. Advances in Mathematics 74, no. 1 (1989), pp. 57–86. [KT-1987] Kazuhiko Koike and Itaru Terada. Young-diagrammatic methods for the representa- tion theory of the classical groups of type Bn, Cn, Dn. Journ...

work page 1989

[9] [9]

Littlewood

[Lit-1958] Dudley E. Littlewood. Products and plethysms of characters with orthogonal, sym- plectic and symmetric groups. Canadian Journal of Mathematics 10 (1958), pp. 17–32. [Mac-1994] A. Murilo Santos Macˆ edo. Universal parametric correlations at the soft edge of the spectrum of random matrix ensembles. Europhysics Letters 26, no. 9 (1994),

work page 1958

[10] [10]

Macdonald

[Mac-1995] Ian G. Macdonald. Symmetric functions and Hall polynomials, 2nd edition. Oxford Mathematical Monographs, Oxford University Press, Oxford,

work page 1995

[11] [11]

Spin q-Whittaker polynomials and de- formed quantum Toda

[MP-2022] Matteo Mucciconi and Leonid Petrov. Spin q-Whittaker polynomials and de- formed quantum Toda. Communications in Mathematical Physics. Volume 389 (2022), pp. 1331–1416 [NOS-2024] Anton Nazarov, Olga Postnova and Travis Scrimshaw. Skew Howe duality and limit shapes of Young diagrams. Journal of the London Mathematical Society 109, no. 1 (2024), e1...

work page 2022

[12] [12]

Infinite wedge and random partitions

[Ok-2001b] Andrei Okounkov. Infinite wedge and random partitions. Selecta Mathematica 7, no. 1 (2001), pp. 57–81. [OP-2006] Andrei Okounkov and Rahul Pandharipande. Gromov-Witten theory, Hurwitz theory, and completed cycles. Annals of Mathematics 163, no. 2 (2006), pp. 517–560. [OR-2003] Andrei Okounkov and Nikolai Reshetikhin. Correlation function of Sch...

work page 2001

[13] [13]

[OV-1996] Grigori Olshanski and Anatoly Vershik

Birkh¨ auser Boston (2006). [OV-1996] Grigori Olshanski and Anatoly Vershik. Ergodic unitarily invariant measures on the space of infinite Hermitian matrices. American Mathematical Society Translation Series 2, no. 175 (1996), pp. 137–176. [PS-2002] Michael Pr¨ ahofer and Herbert Spohn. Scale Invariance of the PNG droplet and the Airy process. Journal of ...

work page 2006

[14] [14]

La correspondance de Robinson

[Sch-1977] Marcel-Paul Sch¨ utzenberger. La correspondance de Robinson. Combinatoire et repr´ esentation du groupe sym´ etrique (1977) pp. 59–113. 56 [Sos-2000] Alexander Soshnikov. Determinantal random point fields. Russian Mathematical Sur- veys 55, no. 5 (2000), pp. 923–975. [Sta-1999] Richard P. Stanley. Enumerative Combinatorics, Vol

work page 1977

[15] [15]

The Cauchy identity for Sp(2n)

[Sun-1990] Sheila Sundaram. The Cauchy identity for Sp(2n). Journal of Combinatorial Theory, Series A 53, no. 2 (1990), pp. 209–238. [Tin-2007] Peter Tingley. Three combinatorial models for ̂sln crystals, with applications to cylin- dric plane partitions. International Mathematics Research Notices, vol. 2007, no. 9, rnm143. [TW-2006] Craig Tracy and Harol...

work page 1990

[16] [16]

Refined Cauchy/Littlewood identities and six-vertex model partition functions: III

[WZJ-2016] Michael Wheeler and Paul Zinn-Justin. Refined Cauchy/Littlewood identities and six-vertex model partition functions: III. Deformed bosons. Advances in Mathematics, Vol. 299 (2016), pp. 543–600. [Whi-1952] Anne M. Whitney. A reduction theorem for totally positive matrices. Journal d’Analyse Math´ ematique 2, no. 1 (1952), pp. 88–92. [Zho-2022] C...

work page 2016