Abstract:
After establishing a geometric Schur-Weyl duality in a general setting, we recall this duality in type A in the finite and affine case. We extend the duality in the affine case to positive parts of the affine algebras. The positive parts have nice ideals coming from geometry, allowing duality for quotients. Some of the quotients of the positive affine Hecke algebra are then identified to some cyclotomic Hecke algebras and the geometric setting allows the construction of canonical bases.

Abstract:
The affine Schur algebra $\widetilde{S}(n,r)$ (of type A) over a field $K$ is defined to be the endomorphism algebra of the tensor space over the extended affine Weyl group of type $A_{r-1}$. By the affine Schur-Weyl duality it is isomorphic to the image of the representation map of the $\mathcal{U}(\hat{\mathfrak{gl}}_{n})$ action on the tensor space when $K$ is the field of complex numbers. We show that $\widetilde{S}(n,r)$ can be defined in another two equivalent ways. Namely, it is the image of the representation map of the semigroup algebra $K\widetilde{GL}_{n,a}$ (defined in Section \ref{S:semigroups}) action on the tensor space and it equals to the 'dual' of a certain formal coalgebra related to this semigroup. By these approaches we can show many relations between different Schur algebras and affine Schur algebras and reprove one side of the affine Schur-Weyl duality.

Abstract:
The quiver Hecke algebra $R$ can be also understood as a generalization of the affine Hecke algebra of type $A$ in the context of the quantum affine Schur-Weyl duality by the results of Kang, Kashiwara and Kim. On the other hand, it is well-known that the Auslander-Reiten(AR) quivers $\Gamma_Q$ of finite simply-laced types have a deep relation with the positive roots systems of the corresponding types. In this paper, we present explicit combinatorial descriptions for the AR-quivers $\Gamma_Q$ of finite type $A$. Using the combinatorial descriptions, we can investigate relations between finite dimensional module categories over the quantum affine algebra $U'_q(A_n^{(i)})$ $(i=1,2)$ and finite dimensional graded module categories over the quiver Hecke algebra $R_{A_n}$ associated to $A_n$ through the generalized quantum affine Schur-Weyl duality functor.

Abstract:
We first provide an explicit combinatorial description of the Auslander-Reiten quiver $\Gamma^Q$ of finite type $D$. Then we can investigate the categories of finite dimensional representations over the quantum affine algebra $U_q'(D^{(i)}_{n+1})$ $(i=1,2)$ and the quiver Hecke algebra $R_{D_{n+1}}$ associated to $D_{n+1}$ $(n \ge 3)$, by using the combinatorial description and the generalized quantum affine Schur-Weyl duality functor. As applications, we can prove that Dorey's rule holds for the category $\Rep(R_{D_{n+1}})$ and prove an interesting difference between multiplicity free positive roots and multiplicity non-free positive roots.

Abstract:
We prove that there is a natural grading-preserving isomorphism of $\sl$-modules between the basic module of the affine Lie algebra $\widehat\sl$ (with the homogeneous grading) and a Schur--Weyl module of the infinite symmetric group $\sinf$ with a grading defined through the combinatorial notion of the major index of a Young tableau, and study the properties of this isomorphism. The results reveal new and deep interrelations between the representation theory of $\widehat\sl$ and the Virasoro algebra on the one hand, and the representation theory of $\sinf$ and the related combinatorics on the other hand.

Abstract:
We study imaginary representations of the Khovanov-Lauda-Rouquier algebras of affine Lie type. Irreducible modules for such algebras arise as simple heads of standard modules. In order to define standard modules one needs to have a cuspidal system for a fixed convex preorder. A cuspidal system consists of irreducible cuspidal modules---one for each real positive root for the corresponding affine root system ${\tt X}_l^{(1)}$, as well as irreducible imaginary modules---one for each $l$-multipartition. We study imaginary modules by means of `imaginary Schur-Weyl duality'. We introduce an imaginary analogue of tensor space and the imaginary Schur algebra. We construct a projective generator for the imaginary Schur algebra, which yields a Morita equivalence between the imaginary and the classical Schur algebra. We construct imaginary analogues of Gelfand-Graev representations, Ringel duality and the Jacobi-Trudy formula.

Abstract:
Complete proofs of Schur-Weyl duality in positive characteristic are scarce in the literature. The purpose of this survey is to write out the details of such a proof, deriving the result in positive characteristic from the classical result in characteristic zero, using only known facts from representation theory.

Abstract:
We introduce new notions on the sequences of positive roots by using Auslander-Reiten quivers. Then we can prove that the new notions provide interesting information on the representation theories of KLR-algebras, quantum groups and quantum affine algebras including generalized Dorey's rule, bases theory for quantum groups, and denominator formulas between fundamental representations.

Abstract:
We review some recent advances in modular representation theory of symmetric groups and related Hecke algebras. We discuss connections with Khovanov-Lauda-Rouquier algebras and gradings on the blocks of the group algebras $F\Sigma_n$, which these connections reveal; graded categorification and connections with quantum groups and crystal bases; modular branching rules and the Mullineaux map; graded cellular structure and graded Specht modules; cuspidal systems for affine KLR algebras and imaginary Schur-Weyl duality, which connects representation theory of these algebras to the usual Schur algebras of smaller rank.

Abstract:
The integral formulae over the unitary group $\unitary{d}$ are reviewed with new results and new proofs. The normalization and the bi-invariance of the uniform Haar measure play the key role for these computations. These facts are based on Schur-Weyl duality, a powerful tool from representation theory of group.