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Cedric Lecouvey

Publications and source records attributed to Cedric Lecouvey.

At least 19 recordsLinked to original sources
arXiv2019

Atomic decomposition of characters and crystals

Lascoux stated that the type A Kostka-Foulkes polynomials K_{lambda,mu}(t) expand positively in terms of so-called atomic polynomials. For any semisimple Lie algebra, the former polynomial is a t-analogue of the multiplicity of the dominant weight mu in the irreducible representation of highest weight lambda. We formulate the atomic decomposition in arbitrary type, and view it as a strengthening of the monotonicity of K_{lambda,mu}(t). We also define a combinatorial version of the atomic decomposition, as a decomposition of a modified crystal graph. We prove that this stronger version holds in type A (which provides a new, conceptual approach to Lascoux's statement), in types B, C, and D in a stable range for t=1, as well as in some other cases, while we conjecture that it holds more generally. Another conjecture stemming from our work leads to an efficient computation of K_{lambda,mu}(t). We also give a geometric interpretation.

math.RT↗
arXiv2017

Combinatorics of generalized exponents

We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type A_{n-1}, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type C_n, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type A_{2n-1}, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig t-analogues associated to zero weight spaces in the irreducible representations of symplectic Lie algebras. We also present three applications of our combinatorial formula, and discuss some implications to relating two type C branching rules. Our methods are expected to extend to the orthogonal types.

math.RT↗
arXiv2017

Central measures on multiplicative graphs, representations of Lie algebras and weight polytopes

To each finite-dimensional representation of a simple Lie algebra is associated a multiplicative graph in the sense of Kerov and Vershik definedfrom the decomposition of its tensor powers into irreducible components. The conditioning of naturalrandom Littelmann paths to stay in their corresponding Weyl chamber is thencontrolled by central measures on this type of graphs. Using the K-theory of associated C*-algebras, Handelman established a homeomorphism between the set of central measures on these multiplicative graphs and the weight polytope of theunderlying representation. In the present paper, we make explicit this homeomorphism independently of Handelman's results by using Littelmann's path model. As a by-product we also get an explicit parametrization of theweight polytope in terms of drifts of random Littelmann paths. This explicit parametrization yields a complete description of harmonic and c-harmonic functions for this Littelmann paths model.

math.RT↗
arXiv2012

Matching subspaces in a field extension

In this paper, we formulate and prove linear analogues of results concerning matchings in groups. A matching in a group G is a bijection f between two finite subsets A,B of G with the property, motivated by old questions on symmetric tensors, that the product af(a)does not belong to A for all a \in A. Necessary and sufficient conditions on G, ensuring the existence of matchings under appropriate hypotheses, are known. Here we consider a similar question in a linear setting. Given a skew field extension K \subset L, where K commutative and central in L, we introduce analogous notions of matchings between finite-dimensional K-subspaces A,B of L, and obtain existence criteria similar to those in the group setting. Our tools mix additive number theory, combinatorics and algebra.

math.NT↗
arXiv2011

Affine crystals, one-dimensional sums and parabolic Lusztig q-analogues

This paper is concerned with one-dimensional sums in classical affine types. We prove a conjecture of the third author and Zabrocki by showing they all decompose in terms of one-dimensional sums related to affine type A provided the rank of the root system considered is sufficiently large. As a consequence, any one-dimensional sum associated to a classical affine root system with sufficiently large rank can be regarded as a parabolic Lusztig q-analogue.

math.QA↗
arXiv2011

X = K under review

We review the X = K conjecture and important ingredients for the proof. We also attach notes on the rank estimate for the X = K theorem to hold and on the strange relation that was found to be valid without the assumption that the rank is sufficiently large. Using the latter one obtains an algorithm to calculate the image of the combinatorial R-matrix and the value of the coenergy function.

math.QA↗
arXiv2008

Lusztig's q-analogue of weight multiplicity and one-dimensional sums for affine root systems

In this paper we complete the proof of the X=K conjecture, that for every family of nonexceptional affine algebras, the graded multiplicities of tensor products of symmetric power Kirillov-Reshetikhin modules known as one-dimensional sums, have a large rank stable limit X that has a simple expression (called the K-polynomial) as nonnegative integer combination of Kostka-Foulkes polynomials. We consider a subfamily of Lusztig's q-analogues of weight multiplicity which we call stable KL polynomials and denote by KL. We give a type-independent proof that K=KL. This proves that X=KL: the family of stable one-dimensional sums coincides with family of stable KL polynomials. Our result generalizes the theorem of Nakayashiki and Yamada which establishes the above equality in the case of one-dimensional sums of affine type A and the Lusztig q-analogue of type A, where both are Kostka-Foulkes polynomials.

math.QA↗
arXiv2008

On q-analogs of weight multiplicities for the Lie superalgebras gl(n,m) and spo(2n,M)

The paper is devoted to the generalization of Lusztig's q-analog of weight multiplicities to the Lie superalgebras gl(n,m) and spo(2n,M). We define such q-analogs K_{lambda,mu}(q) for the typical modules and for the irreducible covariant tensor gl(n,m)-modules of highest weight lambda. For gl(n,m), the defined polynomials have nonnegative integer coefficients if the weight mu is dominant. For spo(2n,M), we show that the positivity property holds when mu is dominant and sufficiently far from a specific wall of the fundamental chamber. We also establish that the q-analog associated to an irreducible covariant tensor gl(n,m)-module of highest weight lambda and a dominant weight mu is the generating series of a simple statistic on the set of semistandard hook-tableaux of shape lambda and weight mu. This statistic can be regarded as a super analog of the charge statistic defined by Lascoux and Schutzenberger.

math.RT↗
arXiv2008

Parabolic Kazhdan-Lusztig polynomials, plethysm and gereralized Hall-Littlewood functions for classical types

We use power sums plethysm operators to introduce H functions which interpolate between the Weyl characters and the Hall-Littlewood functions Q' corresponding to classical Lie groups. The coefficients of these functions on the basis of Weyl characters are parabolic Kazhdan-Lusztig polynomials and thus, are nonnegative. We prove that they can be regarded as quantizations of branching coefficients obtained by restriction to certain Levi subgroups. The H functions associated to linear groups coincide with the polynomials introduced by Lascoux Leclerc and Thibon (LLT polynomials).

math.RT↗
arXiv2008

Stabilized plethysms for the classical Lie groups

The plethysms of the Weyl characters associated to a classical Lie group by the symmetric functions stabilize in large rank. In the case of a power sum plethysm, we prove that the coefficients of the decomposition of this stabilized form on the basis of Weyl characters are branching coefficients which can be determined by a simple algorithm. This generalizes in particular some classical results by Littlewood on the power sum plethysms of Schur functions. We also establish explicit formulas for the outer multiplicities appearing in the decomposition of the tensor square of any irreducible finite dimensional module into its symmetric and antisymmetric parts. These multiplicities can notably be expressed in terms of the Littlewood-Richardson coefficients.

math.RT↗
arXiv2007

Universal graded characters and limit of Lusztig $q$-analogues

Let G be a symplectic or orthogonal complex Lie group with Lie algebra g. As a G-module, the decomposition of the symmetric algebra S(g) into its irreducible components can be explicitely obtained by using identities due to Littlewood. We show that the multiplicities appearing in the decomposition of the k-th graded component of S(g) do not depend on the rank n of g providing n is sufficiently large. Thanks to a classical result by Kostant, we establish a similar result for the k-th graded component of the space H(g) of G-harmonic polynomials. These stabilization properties are equivalent to the existence of a limit in infinitely many variables for the graded characters associated to S(g) and H(g). The limits so obtained are formal series with coefficients in the ring of universal characters introduced by Koike and Terada. From Hesselink expression of the graded character of harmonics, the coefficient of degree k in the Lusztig q-analogue associated to the partition lambda and the weight 0 thus stablizes for n sufficiently large. By using Morris-type recurrence formulas, we prove that this is also true for the polynomials $K_{λ,μ}^{g}(q)$ where mu is a nonempty fixed partition. This can be reformulated in terms of a stability property for the dimension of the components of the Brylinski-Kostant filtration. We also associate to each pair of partitions (lambda, mu) formal series $K_{λ,μ}^{so}(q)$ and $K_{λ,μ}^{sp}(q)$, which can be regarded as natural limit of the Lusztig q-analogues. One gives a duality property for these limits and obtains simple expressions when lambda is a row or a column partition.

math.RT↗
arXiv2006

Matchings in arbitrary groups

A matching in a group G is a bijection f from a subset A to a subset B in G such that af(a) does not belong to A for all a in A. The group G is said to have the matching property if, for any finite subsets A,B in G of same cardinality with B avoiding 1, there is a matching from A to B. Using tools from additive number theory, Losonczy proved a few years ago that the only abelian groups satisfying the matching property are the torsion-free ones and those of prime order. He also proved that, in an abelian group, any finite subset A avoiding 1 admits a matching from A to A. In this paper, we show that both Losonczy's results hold verbatim for arbitrary groups, not only abelian ones. Our main tools are classical theorems of Kemperman and Olson, also pertaining to additive number theory, but specifically developped for possibly nonabelian groups.

math.GR↗
arXiv2006

Signed permutations and the four color theorem

To each permutation $σ$ in $S_{n}$ we associate a triangulation of a fixed $(n+2)$-gon. We then determine the fibers of this association and show that they coincide with the sylvester classes depicted By Novelli, Hivert and Thibon. A signed version of this construction allows us to reformulate the four color theorem in terms of the existence of a signable path between any two permutations in the Cayley graph of the symmetric group $S_{n}.

math.CO↗
arXiv2006

Crystal bases and combinatorics of infinite rank quantum groups

The tensor powers of the vector representation associated to an infinite rank quantum group decompose into irreducible components with multiplicities independant of the infinite root system considered. Although the irreducible modules obtained in this way are not of highest weight, they admit a crystal basis and a canonical basis. This permits in particular to obtain for each familly of classical Lie algebras a Robinson-Schensted correspondence on biwords defined on infinite alphabets. We then depict a structure of bi-crystal on these bi-words. This RS-correspondence yields also a plactic algebra and plactic Schur functions distinct for each infinite root system. Surprisingly, the algebras spanned by these plactic Schur functions are all isomorphic to the algebra of symmetric functions.

math.CO↗
arXiv2004

A duality between $q$-multiplicities in tensor products and $q$-multiplicities of weights for the root systems $B,C$ or $D$

Starting from Jacobi-Trudi's type determinental expressions for the Schur functions corresponding to types $B,C$ and $D,$ we define a natural $q$-analogue of the multiplicity $[V(λ):M(μ)]$ when $M(μ)$ is a tensor product of row or column shaped modules defined by $μ$. We prove that these $q$-multiplicities are equal to certain Kostka-Foulkes polynomials related to the root systems $C$ or $D$. Finally we derive formulas expressing the associated multiplicities in terms of Kostka numbers.

math.RT↗
arXiv2004

Combinatorics of crystal graphs and Kostka-Foulkes polynomials for the root systems $B_{n},C_{n}$ and $D_{n}.$

We use Kashiwara-Nakashima's combinatorics of crystal graphs associated to the roots sytems $B_{n}$ and $D_{n}$ to extend the results of \QCITE{cite}{}{lec3} and \QCITE{cite}{}{Mor} by showing that Morris type recurrence formulas also exist for the orthogonal root systems. We derive from these formulas a statistic on Kashiwara-Nakashima's tableaux of types $B_{n},C_{n}$ and $D_{n}$ generalizing Lascoux-Sch\UNICODE{0xfc}tzenberger's charge and from which it is possible to compute the Kostka-Foulkes polynomials $K_{λ,μ}(q)$ with restrictive conditions on $(λ,μ)$ . This statistic is different from that obtained in \QCITE{cite}{}{lec3} from the cyclage graph structure on tableaux of type $C_{n}$. We show that such a structure also exists for the tableaux of types $B_{n}$ and $D_{n}$ but can not be simply related to the Kostka-Foulkes polynomials. Finally we give explicit formulas for $K_{λ,μ}(q)$ when $| λ| \leq 3,$ or $n=2$ and $μ=0$.

math.CO↗