Journées d'Algèbre

We define a family of symmetric polynomials G_ν,λ(z₁,...,z_n+1,q) indexed by a pair of dominant integral weights for a root system of type A_n. The polynomial G_ν,0(z,q) is the specialized Macdonald polynomial P_ν,0(z,q,0) which is known to be the graded character of a level one Demazure module associated to the affine Lie algebra $\widehat{\mathfrak{sl}}_{n+1}$. We shall see that the G_0,λ(z,q) is the U_q(${\mathfrak{g}}$)–character of the family of prime irreducible representations of quantum affine ${\mathfrak{sl}}_{n+1}$ which arise from the work of Hernandez-Leclerc on monoidal categorification. Under suitable conditions on (ν,λ) (which apply to the pairs (ν,0) and (0,λ)) we prove that G_ν,λ(z,q) is Schur positive, i.e., it can be written as a linear combination of Schur polynomialswith coefficients in $\mathbb{Z}_+[q]$. In fact they also arise as characters of irreducible representations quantum affine ${\mathfrak{sl}}_{n+1}$. A further consequence of our result is an explicit formula for the specialized Macdonald polynomial associated to a non-simply laced Lie algebra as a linear combination of level one Demazure characters thus extending earlier work of Y.Sanderson and B. Ion. The talk is based on joint work with Biswal, Shereen, Wand and also with Brito, Moura. [PDF Slides]

This is a report on joint work with D. Labardini-Fragoso and J. Schröer. We show that for most marked surfaces with non-empty boundary, possibly with punctures, the generic CC-functions form a basis of the corresponding cluster algebras for any choice of geometric coefficients. For surfaces without punctures the τ-reduced components of the corresponding gentle Jacobian algebra are naturally parametrized by X-laminations of the surface, and it is easy to see that for principal coefficients the generic basis coincides with the MSW-bangle basis. [PDF Slides]

Chris Fraser has discovered an action of the extended affine braid group on d strands on the Grassmannian cluster algebra of k-subspaces in n-space, where d is the least common divisor of k and n. We lift this action to the corresponding cluster category first constructed by Geiss-Leclerc-Schroeer in 2008. For this, we use Jensen-King-Su's description of this category as a singularity category in the sense of Buchweitz/Orlov. We conjecture an action of the same braid group on the cluster algebra associated with an arbitrary pair of Dynkin diagrams whose Coxeter numbers are k and n. This is a report on ongoing joint work with Chris Fraser. [French abstract] [Link to the video]

The geometric Satake correspondence can be regarded as a geometric construction of the rational representations of a connected reductive group $G$. In this context, (generalized) Mirković-Vilonen cycles provide bases of tensor products of irreducible representations of $G$ and of the algebra $\mathbb{C}[N]$ of regular functions on the radical unipotent of a Borel subgroup of $G$. We present explicit computations. In particular, we observe that the resulting basis of $\mathbb{C}[N]$ contains many cluster monomials, but differs from Lusztig's dual canonical basis/Kashiwara's upper global basis and from Lusztig's dual semicanonical basis. [PDF Slides]

10 years ago, Hausel, Rodriguez-Villegas and myself gave a conjectural formula for the mixed Hodge polynomial of character varieties (moduli space of representations of fundamental groups of punctured Riemann surfaces with local monodromies in semisimple conjugacy classes of GL(n,C)) in terms of Macdonald symmetric functions. Recently, Mellit proved that our conjecture is true under the specialisation which gives the Poincaré polynomial by giving a geometrical construction of Macdonald symmetric functions. In this talk we will extend our conjecture and results to the case of character varieties associated to the orientation covering of non-orientable compact surfaces (this is a work in progress with Rodriguez-Villegas).

To a complex reflection group W acting on a complex vector space V, and a family of parameters c, one may associate a rational Cherednik algebra. It has a faithful representation on polynomial functions on V, which has a unique simple quotient L known as the spherical module. In joint work with Stephen Griffeth, we determine the support of the spherical module seen as a coherent sheaf on V, depending on c. This is a classical problem which was solved previously by Varagnolo-Vasserot in the Weyl group case with equal parameters, and by Etingof in the Coxeter group case with arbitrary parameters. Our criterion has a very explicit form, in terms of the Schur elements of the Hecke algebra of W and of its parabolic subgroups, if we accept the conjecture that those Hecke algebras are symmetric (this is known for Coxeter groups and for the infinite series, but only for some of the exceptional complex reflections groups). [PDF Slides]

We will review the origins of LLT polynomials and some subsequent developments. [PDF Slides]

Classical sequences of numbers usually lead to interesting q-analogues. The most popular among them are certainly the q-integers and the q-binomial coefficients which both appear in various areas of mathematics and physics. However it seems that q-analogues of rational numbers have not been considered so far. We suggest a definition based on combinatorial properties of rational numbers and continued fractions. The definition of q-rationals naturally extends the one of q-integers and leads to a ratio of polynomials with positive integer coefficients. One can give enumerative interpretations of the coefficients in terms of graphs or quiver representations. There are also links to the Jones polynomials and cluster algebras. Finally the definition of q-rationals extends to a definition for q-real numbers. This is joint work with Valentin Ovsienko.[PDF Slides]

Gentle algebras are a class of associative algebras that arise naturally from dissections of surfaces. They are tame and derived tame, and their class is closed under derived equivalence. In this talk, I will present a numerical invariant that completely determines the derived-equivalence class of a gentle algebra. It is defined using the geometry of the dissected surface from which a gentle algebra arises. This invariant completes the earlier one obtained combinatorially by C.Geiss and D.Avella-Alaminos. (This is a joint work with Claire Amiot and Sibylle Schroll). [Link to the video]

We have defined new invariants for the pair of modules over quantum affine algebras U′_q(${\mathfrak{g}}$) by analyzing their associated R-matrices. It behaves similarly to the invariants Λ for quiver Hecke algebras, which is deeply connected with graded structure of quiver Hecke algebras. With the help of new invariants, we can give a criteria for a category of U′_q(${\mathfrak{g}}$)--modules to become a monoidal categorification of a cluster algebra. This is a joint work with Myungho Kim, Se-jin Oh and Euiyong Park.

The Grothendieck ring K(C) of a category C of finite-dimensional representations of a quantum affine algebra admit natural quantum deformations obtained from perverse sheaves on quiver varieties, from deformed W-algebras or from vertex operators on quantum Heisenberg algebras. We review and discuss the interplay with the cluster algebra structures introduced with Bernard Leclerc as well as with the intricate structure of the category C (based in part on joint works with Bernard Leclerc, with Hironori Oya and with Laura Fedele).

Opper Plamondon and Schroll described a geometric model for the derived category of gentle algebras. In this talk I will explain how to use this model to get one for the derived category of skew-gentle algebras. The main tool is the use of the skew-group algebra structure of the skew-gentle algebras. This is a collaboration with Thomas Brüstle. [PDF Slides]