The geometric construction of DAHA by the equivariant K-theory of the Steinberg-type variety for an affine flag variety by Vasserot, Varagnolo-Vasserot is a precursor of the Coulomb branch construction. It has been generalized to versions of DAHA, such as cyclotomic DAHA naturally in view of Coulomb branches. I would like to recall these results, and then add one new example which seems not be known before.
I will present some recent works with Gunningham, Safronov, Vazi-rani, and Yang (in various combinations) and which compute GLN-, SLN- and PGLN-skein modules for the 3-torus T 3, and related work of Kinnear which generalizes this to mapping tori T 2 ×γ S1, for γ ∈ SL2(ℤ).
The proofs for GLN and SLN start with a description of the skein category of T 2 via the representation theory of double affine Hecke algebras, while for PGLN they rely on an instance of electric-magnetic duality.
In the last decades, there has been an increasing interest in the classification of isomorphism classes of group gradings on a given algebra. We discuss some difficulties concerning the study of group gradings on infinite-dimensional algebras. Then, we present our results on the classification of the gradings on the infinite-dimensional triangular algebra.
The classical shuffle theorem states that the Frobenius character of the space of diagonal harmonics is given by a certain combinatorial sum indexed by parking functions on square lattice paths. The rational shuffle theorem, conjectured by Gorsky-Negut and proven by Mellit, states that the geometric action on symmetric functions (described by Schiffmmann-Vasserot) of certain elliptic Hall algebra elements P(m,n) yield the bigraded Frobenius character of a certain Sn representation. This character is known as the Hikita polynomial. In this talk I will introduce the higher-rank rational (q,t)-Catalan polynomials and show these are equal to finite truncations of the Hikita polynomial. By generalizing results of Gorsky-Mazin-Vazirani and constructing an explicit bijection between rational semistandard parking functions and affine compositions, I will derive a finite analogue of the rational shuffle theorem in the context of spherical double affine Hecke algebras.
The affine Hecke algebra has a remarkable commutative subalgebra corresponding to the coroot lattice in the affine Weyl group. Its nature is encoded in the Bernstein presentation and reveals important representation-theoretic properties of the algebra. If one considers categorifications of the Hecke algebra, for instance the diagrammatic category, the above subalgebra corresponds to a class of complexes in the homotopy category called Wakimoto sheaves, which can be seen as Rouquier complexes. In this talk I will introduce the affine Hecke algebra, the diagrammatic category and the objects mentioned above. Then I will describe some reduced representarives for Rouquier complexes and present some results about the extension groups between Wakimoto sheaves in affine type A1.
The structure of small cancellation groups is well known. Тhey are widely used in construction of groups with unusual properties (for example Burnside groups and Tarskii monster). We were interested in developing a similar theory for rings. However, such theory meets significant difficulties because, unlike groups, rings have two operations: addition and multiplication. I will speak about small cancellation conditions for rings that we introduced. These conditions provide the desired properties. I will discuss our way towards these conditions, examples and possible applications of small cancellation rings.
Upper triangular, and more generally, block-triangular matrices, are rather important in linear algebra, and also in ring theory, namely in the theory of PI algebras. The group gradings on such algebras have been studied extensively during the last decades. In 2007 A. Valenti and M. Zaicev conjectured that every grading on these algebras is obtained from an elementary grading on a block-triangular matrix algebra and a division grading on a matrix algebra. In this talk we present recent results on this problem.
We present recent developments in symplectic geometry and explain how they motivated new results in the study of cluster algebras. First, we introduce a geometric problem: the study of Lagrangian surfaces in the standard symplectic 4-ball bounding Legendrian knots in the standard contact 3-sphere. Thanks to results from the microlocal theory of sheaves, which we will survey, we then show that this geometric problem gives rise to an interesting moduli space. In fact, we establish a bridge translating geometric operations, such as Lagrangian disk surgeries, into algebraic properties of this moduli space, such as the existence of cluster algebra structures. The talk is intended for a broad symplectic audience and all key ideas will be introduced and motivated.
In this talk, we deal with varieties of PI-superalgebras with graded involution of finite basic rank over a field of characteristic zero and we present some recent results concerning the minimality of these varieties (of fixed *-graded exponent) and the factorability of their *-graded polynomial identities.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
KLR algebras of type A have been a revolution in the representation theory of Hecke algebras of a type A flavour, thanks to the the Brundan-Kleshchev-Rouquier isomorphism relating them explicitly to the affine Hecke algebra of type A. KLR algebras of other types exist but are not related to affine Hecke algebras of other types. In this talk I will present a generalisation of the KLR presentation for the affine Hecke algebra of type B and I will discuss some applications.
