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Jean-Yves Thibon

Publications and source records attributed to Jean-Yves Thibon.

At least 19 recordsLinked to original sources
arXiv2026

Schur positivity of the spiders $S(a,2,1)$ and $S(a,4,1)$ via noncommutative symmetric functions

We prove that the spider graphs $S(a,2,1)$ and $S(a,4,1)$ are Schur positive for all integers $a\ge1$. Together with the known $e$-positivity results, this completes the $e$- and Schur-positivity classification of both families. Our approach uses noncommutative symmetric functions, including a particularly simple ribbon expansion for the path lift with coefficients given by powers of two. We give a new proof of the Shareshian--Wachs path formula at $t=1$ and construct corresponding lifts for spiders. The Littlewood--Richardson rule converts their ribbon expansions into a general Schur-coefficient formula in terms of weighted Yamanouchi words. Mass-preserving multi-injections and a reduction to finitely many inequalities in degree $10$ then prove the required positivity; exact computer verification of these inequalities completes the proof in full generality.

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arXiv2026

Jack Content Operators and the Deformed ${\mathcal W}_{1+\infty}$ Algebra

Frenkel and Wang obtained a representation of the Virasoro algebra by commuting Goulden's cut-and-join operator with the Heisenberg generators. A vertex-operator construction by Lascoux and the author extends this representation to $\mathcal W_{1+\infty}$ by means of differential operators whose eigenvalues are the power sums of the contents of a Young diagram. We develop a Jack deformation in the spherical degenerate double affine Hecke algebra and its stable limit. Starting from the Heckman--Polychronakos integrals, we isolate operators whose eigenvalues are the power sums of the $α$-contents. Because the Goulden--Jackson product is defined in the convention dual to the usual Calogero--Sutherland Hamiltonians, the multiplication operators $Δ_μ(α)$ are obtained by taking Hall adjoints. This gives conceptual derivations of the Jack cut-and-join operator and of the stable $3$-cycle operator. A normal-ordering construction due to Sergeev and Veselov makes the latter calculation explicit and suggests an integral form over $\mathbb Z[α]$. The commutators of the cut-and-join operator contain one half of the usual Feigin--Fuchs realization, but this Virasoro completion is not the deformation of the Frenkel--Wang construction. The latter takes place in the deformed $\mathcal W_{1+\infty}$ algebra $\mathbf{SH}^c$, equivalently in the affine Yangian of $\mathfrak{gl}_1$: in our normalization its first nontrivial Cartan mode is $ψ_3=3Δ_2(α)+2(α-1)E$, and its commutators with the first raising and lowering modes recursively generate the remaining currents. At $α=1$ these relations specialize to the central-charge-one $\mathcal W_{1+\infty}$ representation used by Lascoux and the author.

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arXiv2026

Shifted Macdonald Polynomials and the $(q,t)$-Deformed Goulden--Jackson Product

In a preceding article, we introduced stable symmetric series encoding simultaneously the normalized conjugacy classes of all symmetric groups. The same rational series remain stable for the Jack-deformed Goulden--Jackson product. We investigate their two-parameter Macdonald analogue. Starting from the $(q,t)$-deformed class product (dual to the coproduct diagonal on the $J$-basis), we construct the unique infinite series whose multiplication realizes any shifted Macdonald eigenvalue. In contrast with the classical and Jack cases, this transform is no longer multiplication by a fixed explicit series. We identify it with a composition of a Cauchy multiplication, the integral nabla operator, and a simple diagonal operator. We then determine the series realizing the Nazarov--Sklyanin operators $A^{(k)}$, derive a single generating series for all column partitions, and compare our construction with the Macdonald characters and Theta operators of Ben Dali and D'Adderio.

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arXiv2026

Stable Symmetric Series, Differential Operators, and Jack Deformations

We introduce stable symmetric series which encode normalized conjugacy classes and their multiplication operators simultaneously for all symmetric groups. This gives a direct route from the Ivanov--Kerov algebra to shifted symmetric functions and to differential operators in $U(\mathcal W_{1+\infty})$. Using the Goulden--Jackson product, we extend the construction to Jack polynomials, recover shifted Jack eigenvalues and Pieri-type relations, and obtain explicit candidate operators in degrees three and four. Their real and quaternionic specializations to zonal polynomials are verified by Gaussian matrix integrals and exhaustive Wick enumeration.

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arXiv2026

The noncommutative geode

We investigate the geode and some of its generalizations from the point of view on noncommutative symmetric functions.

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arXiv2025

Noncommutative chromatic quasi-symmetric functions, Macdonald polynomials, and the Yang-Baxter equation

As shown in our paper [JCTA 177 (2021), Paper No. 105305], the chromatic quasi-symmetric function of Shareshian-Wachs can be lifted to ${\bf WQSym}$, the algebra of quasi-symmetric functions in noncommuting variables. We investigate here its behaviour with respect to classical transformations of alphabets and propose a noncommutative analogue of Macdonald polynomials compatible with a noncommutative version of the Haglund-Wilson formula. We also introduce a multi-$t$ version of these noncommutative analogues. For rectangular partitions, their commutative images at $q=0$ appear to coincide with the multi-$t$ Hall-Littlewood functions introduced in [Lett. Math. Phys. 35 (1995), 359]. This leads us to conjecture that for rectangular partitions, multi-$t$ Macdonald polynomials are obtained as equivariant traces of certain Yang-Baxter elements of Hecke algebras. We also conjecture that all (ordinary) Macdonald polynomials can be obtained in this way. We conclude with some remarks relating various aspects of quasi-symmetric chromatic functions to calculations in Hecke algebras. In particular, we show that all modular relations are given by the product formula of the Kazhdan-Lusztig basis.

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arXiv2024

Acyclic orientations and Hessenberg varieties

We exhibit a bijection between acyclic orientations of a Dyck graph and Tymoczko cells of a regular nilpotent Hessenberg variety. This implies the Shareshian-Wachs formula for the sum of the coefficients of the chromatic quasi-symmetric function of a Dyck graph in the elementary basis.

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arXiv2023

Stability properties of inner plethyms (Lecture Notes)

The inner plethysm of symmetric functions corresponds to the $λ$-ring operations of the representation ring $R({\mathfrak S}_n)$ of the symmetric group. It is known since the work of Littlewood that this operation possesses stability properties w.r.t. $n$. These properties have been explained in terms of vertex operators [Scharf and Thibon, Adv. Math. 104 (1994), 30-58]. Another approach [Orellana and Zabrocki, Adv. Math. 390 (2021), \# 107943], based on an expression of character values as symmetric functions of the eigenvalues of permutation matrices, has been proposed recently. This note develops the theory from scratch, discusses the link between both approaches and provides new proofs of some recent results.

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arXiv2023

Tree expansions of some Lie idempotents}

We prove that the Catalan Lie idempotent $D_n(a,b)$, introduced in [Menous {\it et al.}, Adv. Appl. Math. 51 (2013), 177] can be refined by introducing $n$ independent parameters $a_0,\ldots,a_{n-1}$ and that the coefficient of each monomial is itself a Lie idempotent in the descent algebra. These new idempotents are multiplicity-free sums of subsets of the Poincaré-Birkhoff-Witt basis of the Lie module. These results are obtained by embedding noncommutative symmetric functions into the dual noncommutative Connes-Kreimer algebra, which also allows us to interpret, and rederive in a simpler way, Chapoton's results on a two-parameter tree expanded series.

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arXiv2021

Pinnacle sets revisited

In 2017, Davis, Nelson, Petersen, and Tenner [Discrete Math. 341 (2018),3249--3270] initiated the combinatorics of pinnacles in permutations. We provide a simple and efficient recursion to compute $p_n(S)$, the number of permutations of $S_n$ with pinnacle set $S$, and a conjectural closed formula for the related numbers $q_n(S)$. We determine the lexicographically minimal elements of the orbits of the modified Foata-Strehl action, prove that these elements form a lower ideal of the left weak order and characterize and count the maximal elements of this ideal.

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arXiv2020

Combinatorial Hopf algebras in noncommutative probabilility

We prove that the generalized moment-cumulant relations introduced in [arXiv:1711.00219] are given by the action of the Eulerian idempotents on the Solomon-Tits algebras, whose direct sum builds up the Hopf algebra of Word Quasi-Symmetric Functions $\WQSym$. We prove $t$-analogues of these identities (in which the coefficient of $t$ gives back the original version), and a similar $t$-analogue of Goldberg's formula for the coefficients of the Hausdorff series. This amounts to the determination of the action of all the Eulerian idempotents on a product of exponentials.

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arXiv2020

Quadri-algebras, preLie algebras, and the Catalan family of Lie idempotents

We compute the expansion of the Catalan family of Lie idempotents introduced in [Menous et al., Adv. Applied Math. 51 (2013), 177-22] on the PBW basis of the Lie module. It is found that the coefficient of a tree depends only on its number of left and right internal edges. In particular, the Catalan idempotents belong to a preLie algebra based on naked binary trees, of which we identify several Lie and preLie subalgebras.

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arXiv2020

Duplicial algebras and Lagrange inversion

We provide operadic interpretations for two Hopf subalgebras of the algebra of parking functions. The Catalan subalgebra is identified with the free duplicial algebra on one generator, and the Schröder subalgebra is interpreted by means of a new operad, which we call triduplicial. The noncommutative Lagrange inversion formula is then interpreted in terms of duplicial structures. The generic solution of the noncommutative inversion problem appears as the formal sum of all parking functions. This suggests that combinatorial generating functions derived by functional inversion should be obtainable by evaluating a suitable character on this generic solution. This idea is illustrated by means of the Narayana polynomials, of which we obtain bivariate "super-analogues" by lifting to parking functions a classical character of the algebra of symmetric functions. Other characters, such as evaluation of symmetric functions on a binomial element, are also discussed.

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arXiv2020

Noncommutative unicellular LLT polynomials

It is known that unicellular LLT polynomials are related to the quasi-symmetric chromatic polynomials of certain graphs by the $(t-1)$-transform of symmetric functions. We investigate the extension of this transformation to various combinatorial Hopf algebras and prove a noncommutative version of this property.

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arXiv2019

The Hopf algebras of signed permutations, of weak quasi-symmetric functions and of Malvenuto-Reutenauer

This paper builds on two covering Hopf algebras of the Hopf algebra QSym of quasi-symmetric functions, with linear bases parameterized by compositions. One is the Malvenuto-Reutenauer Hopf algebra SSym of permutations, mapped onto QSym by taking descents of permutations. The other one is the recently introduced Hopf algebra RQSym of weak quasi-symmetric functions, mapped onto QSym by extracting compositions from weak compositions. We extend these two surjective Hopf algebra homomorphisms into a commutative diagram by introducing a Hopf algebra HSym, linearly spanned by signed permutations from the hyperoctahedral groups, equipped with the shifted quasi-shuffle product and deconcatenation coproduct. Extracting a permutation from a signed permutation defines a Hopf algebra surjection form HSym to SSym and taking a suitable descent from a signed permutation defines a linear surjection from HSym to RQSym. The notion of signed $P$-partitions from signed permutations is introduced which, by taking generating functions, gives fundamental weak quasi-symmetric functions and sends the shifted quasi-shuffle product to the product of the corresponding generating functions. Together with the existing Hopf algebra surjections from SSym and RQSym to QSym, we obtain a commutative diagram of Hopf algebras revealing the close relationship among compositions, weak compositions, permutations and signed permutations.

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arXiv2019

A noncommutative cycle index and new bases of quasi-symmetric functions and noncommutative symmetric functions

We define a new basis of the algebra of quasi-symmetric functions by lifting the cycle-index polynomials of symmetric groups to noncommutative polynomials with coefficients in the algebra of free quasi-symmetric functions, and then projecting the coefficients to $QSym$. By duality, we obtain a basis of noncommutative symmetric functions, for which a product formula and a recurrence in the form of a combinatorial complex are obtained. This basis allows to identify noncommutative symmetric functions with the quotient of FQSym induced by the pattern-replacement relation $321 \equiv 231$ and $312 \equiv 132$.

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