Cédric Lecouvey (Université de Tours): Crystal graph theory and some of its generalizations
Jeremy Lovejoy (Université de Paris): Bailey pairs in combinatorics, number theory, and knot theory
Florian Aigner (University of Vienna): Alternating sign matrices and totally symmetric plane partitions
We present a new family of Schur positive symmetric polynomials, which are defined as sums over totally symmetric plane partitions. In the first part of the talk, we show that these symmetric polynomials generalise a multivariate generating function of alternating sign matrices. This establishes a new connection between ASMs and a class of plane partitions, thereby complementing the fact that ASMs are equinumerous with totally symmetric self-complementary plane partitions as well as with descending plane partitions. In the second part, we relate three specialisations of these symmetric polynomials to a weighted enumeration of certain well-known classes of column strict shifted plane partitions that generalise descending plane partitions. This is based on joint work with Ilse Fischer and Matjaž Konvalinka, Philippe Nadeau and Vasu Tewari.
Seamus Albion (University of Vienna): 2-core Littlewood identities
In his 1940 textbook on group characters, Littlewood wrote down two summation formulae for Schur functions where the sum is restricted to partitions whose Young diagram contains only columns or rows of even length. Now called Littlewood identities, these formulae and their generalisations have found many applications in combinatorics, representation theory and hypergeometric series. Recently Lee, Rains and Warnaar conjectured a pair of novel Littlewood-type identities for Macdonald polynomials where the sum is over partitions with empty 2-core. I will explain how to resolve their conjectures in the Schur function case, and discuss some further fascinating conjectures involving partitions with empty 2-core.
Olga Azenhas (Universidade de Coimbra ): Symplectic cactus action on crystals of Kashiwara-Nakashima tableaux - Part I
Halacheva has defined the cactus group J_g for any finite dimensional complex reductive Lie algebra g, considering connected sub-diagrams of the Dynkin diagram of g. Then she showed that there is an internal action of J_g on the corresponding g-crystal via partial Sch\"utzenberger-Lusztig involutions. We realize explicitly the action of the symplectic cactus group on crystals of Kashiwara-Nakashima tableaux. Our realization of the partial Sch\"utzenberger-Lusztig involutions uses the Sheats-Lecouvey symplectic jeu de taquin, reversal and Baker virtualization. As an application, we define a symplectic Berenstein-Kirilov group and show that it is a quotient of the symplectic cactus group. Joint work with M. Tarighat and J. Torres.
Bérénice Delcroix-Oger (Université de Paris): Tridendriform structures on the faces of hypergraph associahedra
In 2004, Jean-Louis Loday and Maria Ronco introduced a splitting of some associative products in three parts and called these products tridendriform product. A toy example for this product is obtained by considering the shuffle product on planar trees and splitting this product according to the tree to which the root originally belongs. Emily Burgunder and Maria Ronco then extended their definition to surjections in 2010. Planar trees and surjections are both faces of two well-known polytopes: associahedra and permutohedra. In 2006, Michael Carr and Satyan Devadoss introduced a general frame containing associahedra and permutohedra : graph associahedra. In 2020, Maria Ronco defined a general associative products on graph associahedra. In ... Kosta Dosen et Zoran Petric introduced a new description of Postnikov's nestohedra in terms of hypergraph associahedra. In a recent work with Pierre-Louis Curien and Jovana Obradovic, we extend Ronco's construction to hypergraph associahedra and introduce a tridendriform splitting of it. After a presentation of the context and preliminary definitions, I will present this construction and some examples.
Hans Höngesberg (University of Vienna): Alternating sign matrices with reflective symmetry and plane partitions: $n+3$ pairs of equivalent statistics
Vertically symmetric alternating sign matrices are known to be equinumerous with lozenge tilings of a hexagon with a central triangular hole of size $2$ that exhibit a cyclical as well as a vertical symmetry, but no bijection between these two classes of objects has been constructed so far. To approach a possible bijection, we introduce $n+3$ parameters for both objects and show that the joint distributions coincide. In fact, we present several versions of such results, but in all cases certain natural extensions of the original objects are necessary and that may hint at why it is so hard to come up with an explicit bijection. This is joint work with Ilse Fischer.
Isaac Konan (Université Lyon 1): The combinatorics of (k,l)-lecture hall partitions
In 1997, Bousquet-Mélou and Eriksson stated a broad generalization of Euler's distinct-odd partition theorem, namely the $(k,l)$-Euler theorem. Their identity involved the $(k,l)$-lecture-hall partitions, which, unlike usual difference conditions of partitions in Rogers-Ramanujan type identities, satisfy some ratio constraints. In a 2008 paper, in response to a question suggested by Richard Stanley, Savage and Yee provided a simple bijection for the $l$-lecture-hall partitions (the case $k=l$), whose specialization in $l=2$ corresponds to Sylvester's bijection. Subsequently, as an open question, a generalization of their bijection was suggested for the case $k,l\geq 2$. In this talk, we provide and sketch a proof of slight variations of the suggested bijection, not only for the case $k,l\geq 2$, but also for the cases $(k,1)$ and $(1,k)$ with $k\geq 4$. Furthermore, if time allows, we will show that our bijections equal the recursive bijections given by Bousquet-M\'elou and Eriksson in their recursive proof of the $(k,l)$-lecture hall theorem and provide the analogous recursive bijection for the $(k,l)$-Euler theorem.
Josef Küstner (University of Vienna): Lattice paths and negatively indexed weight-dependent binomial coefficients
In 1992, Daniel Loeb considered a natural extension of the binomial coefficients to negative entries and gave a combinatorial interpretation in terms of hybrid sets. He showed that many of the fundamental properties of binomial coefficients continue to hold in this extended setting. Recently, Sam Formichella and Armin Straub showed that these results can be extended to the $q$-binomial coefficients with arbitrary integer values and extended the work of Loeb further by examining arithmetic properties of the $q$-binomial coefficients. In our work, we give an alternative combinatorial interpretation in terms of lattice paths and consider an extension of the more general weight-dependent binomial coefficients, first defined by Schlosser, to arbitrary integer values. Remarkably, many of the results of Loeb, Formichella and Straub continue to hold in the general weighted setting. In this talk I will also examine some important special cases of the weight-dependent binomial coefficients, including ordinary, $q$- and elliptic binomial coefficients as well as elementary and complete homogeneous symmetric functions. This is joint work with Michael Schlosser and Meesue Yoo.
Yvan Le Borgne (Université Bordeaux 1): On periodic configurations for sandpile model on square lattice
The sandpile model is a discrete model for diffusion well defined on any finite graph where a vertex is distinguished and called the sink. Recurrent configurations are central in this model, in bijection with spanning trees and characterized by an algorithmic criterion due to Dhar. We will present a weaker version of this criterion that seems suited to define an analogue notion of periodic recurrent configuration on the (infinite) square lattice. It will lead to a notion of Tutte polynomial parametrized by the direction in which the sink is sent to infinity. This gives several bi-statistics with constant marginal statistics and symmetric by planar duality within a framework where on complete graphs less obviously symmetric bi-statistic, some famous q,t-Catalan may be defined. (joint work with Henri Derycke partially presented in "Restricted Tutte polynomials for some periodic oriented forests on infinite square lattice" FPSAC 2019, https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2019/51.html)
Matteo Mucciconi (University of Warwick): Bijective proof of Cauchy Identities for q-Whittaker polynomials
Vu Nguyen Dinh (Université Sorbonne Paris Nord): Drinfeld-Kohno Lie algebra and Lazard elimination
In this talk, we present a relation between the decomposition of Drinfeld-Kohno Lie algebra and the Lazard elimination. This paves the way for finding all solutions of KZ_n, for n=3,4.
Jiayue Qi (RISC): Maker-breaker domination number
We study the Maker--Breaker domination game played by Dominator and Staller on the vertex set of a given graph. Dominator wins when the vertices he has claimed form a dominating set of the graph. Staller wins if she makes it impossible for Dominator to win, or equivalently, she is able to claim some vertex and all its neighbours. Maker--Breaker domination number $\gamma_{MB}(G)$ ($\gamma '_{MB}(G)$) of a graph $G$ is defined to be the minimum number of moves for Dominator to guarantee his winning when he plays first (second). We give a structural characterization of the graphs $G$ with $\gamma (G) =\gamma _{MB}(G)= 2$. We give upper bounds for the Maker--Breaker domination number of the Cartesian product of the complete graph with two vertices and an arbitrary graph. Most importantly, we prove that $\gamma _{MB}(P_2\square P_n)=n-2$ for $n\geq 13$ and $\gamma'_{MB}(P_2\square P_n)=n$ for $n\geq 1$.
Salim Rostam (ENS Rennes): Asymptotics of Young diagrams
A partition is a finite sequence of positive integers. The Plancherel (probability) measure on the set of partitions comes from the representation theory of the symmetric group. We will introduce two aspects of partition theory: the fact that a certain associated process on $\mathbb{Z}$ is determinantal under the Plancherel measure, and the (representation-theoretic) notion of core of a partition. We will then show how to obtain the asymptotic size of the core of a partition under the Plancherel measure.
Nat Thiem (University of Colorado Boulder): Categorifying combinatorial Hopf algebras
The idea of organizing infinite families of combinatorial objects into graded vector spaces with an inductive structure (realized as a Hopf algebra) is a fundamental idea in algebraic combinatorics. However, one quickly encounters the problem of too much flexibility: what are the good bases?, what is the “right” inner product structure?, what are the best algebra morphisms on the space?, etc. One approach to making such choices more canonical is to associate the combinatorial Hopf algebra to the representation theory of some family of algebras (or, in our case, a tower of groups). This talk explores a combinatorial approach to making such a connection with a special emphasis on the case of noncommutative symmetric functions, where we obtain a family of such correspondences. This is joint work with F. Aliniaeifard.
Jacinta Torres (Institute of Mathematics of the Polish Academy of Sciences): Minuscule exceptional Schubert varieties
I will talk about Schubert varieties in the homogeneous spaces G/P, where G is an exceptional reductive algebraic group and P is a minuscule parabolic subgroup. By computational methods as well as hands-on analysis, we explicitly describe the defining ideals of the intersections of these Schubert varieties with the big open cell. We show that some of them as well as their resolutions are related to well-known ideals resp. resolutions in other contexts. This is joint work with Sara Angela Filippini and Jerzy Weyman.
David Wahiche (Université Lyon 1): A q-analogue of Nekrasov-Okounkov for type C
Between 2006 and 2008, using various methods coming from representation theory, gauge theory, combinatorics, several authors proved the so-called Nekrasov-Okounkov formula involving hook-length of integer partitions. Later Dehaye and Han proved an identity which can be reformulated as a q-analogue of Nekrasov-Okounkov identity. This result was generalized by both Rains-Warnaar and Carlsson-Rodriguez Villegas in 2018. In this talk, I will explain how we can use the Littlewood decomposition on partitions and its interpretation in terms of bi-infinite words to derive a q-analogue of Pétréolle's Nekrasov-Okounkov type formula for double distinct partitions.