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Register a new userLectures on dualities ABC in representation theory
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In these lecture notes for a summer mini-course, we provide an exposition on quantum groups and Hecke algebras, including (quasi) R-matrix, canonical basis, and $q$-Schur duality. Then we formulate their counterparts in the setting of $imath$quantum groups arising from quantum symmetric pairs, including (quasi) K-matrix, $imath$-canonical basis, and $imath$Schur duality. As an application, the ($imath$-)canonical bases are used to formulate Kazhdan-Lusztig theories and character formulas in the BGG categories for Lie (super)algebras of type A-D. Finally, geometric constructions for $q$-Schur and $imath$Schur dualities are provided.
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Structures in low-dimensional topology and low-dimensional geometry -- often combined with ideas from (quantum) field theory -- can explain and inspire concepts in algebra and in representation theory and their categorifie
These myh lectures at the Park City conference in 1998.
There is both theoretical and numerical evidence that the set of irreducible representations of a reductive group over local or finite fields is naturally partitioned into families according to analytic properties of representations. Examples of such properties are the rate of decay at infinity of matrix coefficients in the local field setting, and the order of magnitude of character ratios in the finite field situation. In these notes we describe known results, new results, and conjectures in the theory of size of representations of classical groups over finite fields, whose ultimate goal is to classify the above mentioned families of representations and accordingly to estimate the relevant analytic properties of each family. Specifically, we treat two main issues: the first is the introduction of a rigorous definition of a notion of size for representations of classical groups, and the second issue is a method to construct and obtain information on each family of representation of a given size. In particular, we propose several compatible notions of size that we call U-RANK, TENSOR RANK and ASYMPTOTIC RANK, and we develop a method called ETA CORRESPONDENCE to construct the families of representation of each given rank. Rank suggests a new way to organize the representations of classical groups over finite and local fields - a way in which the building blocks are the smallest representations. This is in contrast to Harish-Chandras philosophy of cusp forms that is the main organizational principle since the 60s, and in it the building blocks are the cuspidal representations which are, in some sense, the LARGEST.
These are lecture notes written at the University of Zurich during spring 2014 and spring 2015. The first part of the notes gives an introduction to probability theory. It explains the notion of random events and random variables, probability measures, expectation, distributions, characteristic function, independence of random variables, types of convergence and limit theorems. The first part is separated into two different chapters. The first chapter is about combinatorial aspects of probability theory and the second chapter is the actual introduction to probability theory, which contains the modern probability language. The second part covers conditional expectations, martingales and Markov chains, which are easily accessible after reading the first part. The chapters are exactly covered in this order and go into some more details of the respective topic.
Let $U_q(mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $mathcal{C}_{mathfrak{g}}$ be Hernandez-Leclercs category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in $mathcal{C}_{mathfrak{g}}$. We introduce the notion of a strong (complete) duality datum $D$ and prove that, when $D$ is strong, the induced duality functor $F_D$ sends simple modules to simple modules and preserves the invariants $Lambda$ and $Lambda^infty$ introduced by the authors. We next define the reflections $mathcal{S}_k$ and $mathcal{S}^{-1}_k$ acting on strong duality data $D$. We prove that if $D$ is a strong (resp. complete) duality datum, then $mathcal{S}_k(D)$ and $mathcal{S}_k^{-1}(D)$ are also strong (resp. complete ) duality data. We finally introduce the notion of affine cuspidal modules in $mathcal{C}_{mathfrak{g}}$ by using the duality functor $F_D$, and develop the cuspidal module theory for quantum affine algebras similarly to the quiver Hecke algebra case.
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