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Let H be a pointed Hopf algebra with abelian coradical. Let A\supseteq B be left (or right) coideal subalgebras of H that contain the coradical of H. We show that A has a PBW basis over B, provided that H satisfies certain mild conditions. In the case that H is a connected graded Hopf algebra of characteristic zero and A and B are both homogeneous of finite Gelfand-Kirillov dimension, we show that A is a graded iterated Ore extension of B. These results turn out to be conceptual consequences of a structure theorem for each pair S\supseteq T of homogeneous coideal subalgebras of a connected graded braided bialgebra R with braiding satisfying certain mild conditions. The structure theorem claims the existence of a well-behaved PBW basis of S over T. The approach to the structure theorem is constructive by means of a combinatorial method based on Lyndon words and braided commutators, which is originally developed by V. K. Kharchenko [Algebra Log. 38 (1999), pp. 476–507, 509] for primitively generated braided Hopf algebras of diagonal type. Since in our context we don't priorilly assume R to be primitively generated, new methods and ideas are introduced to handle the corresponding difficulties, among others. [ABSTRACT FROM AUTHOR]

We prove that large classes of algebras in the framework of root of unity quantum cluster algebras have the structures of maximal orders in central simple algebras and Cayley–Hamilton algebras in the sense of Procesi. We show that every root of unity upper quantum cluster algebra is a maximal order and obtain an explicit formula for its reduced trace. Under mild assumptions, inside each such algebra we construct a canonical central subalgebra isomorphic to the underlying upper cluster algebra, such that the pair is a Cayley–Hamilton algebra; its fully Azumaya locus is shown to contain a copy of the underlying cluster \mathcal {A}-variety. Both results are proved in the wider generality of intersections of mixed quantum tori over subcollections of seeds. Furthermore, we prove that all monomial subalgebras of root of unity quantum tori are Cayley–Hamilton algebras and classify those ones that are maximal orders. Arbitrary intersections of those over subsets of seeds are also proved to be Cayley–Hamilton algebras. Previous approaches to constructing maximal orders relied on filtration and homological methods. We use new methods based on cluster algebras. [ABSTRACT FROM AUTHOR]

We give a construction of generalized cluster varieties and generalized cluster scattering diagrams for reciprocal generalized cluster algebras, the latter of which were defined by Chekhov and Shapiro. These constructions are analogous to the structures given for ordinary cluster algebras in the work of Gross, Hacking, Keel, and Kontsevich. As a consequence of these constructions, we are also able to construct theta functions for generalized cluster algebras, again in the reciprocal case, and demonstrate a number of their structural properties. [ABSTRACT FROM AUTHOR]

A nilpotent endomorphism of a quiver representation induces a linear transformation on the vector space at each vertex. Generically among all nilpotent endomorphisms, there is a well-defined Jordan form for these linear transformations, which is an interesting new invariant of a quiver representation. If Q is a Dynkin quiver and m is a minuscule vertex, we show that representations consisting of direct sums of indecomposable representations all including m in their support, the category of which we denote by C Q , m , are determined up to isomorphism by this invariant. We use this invariant to define a bijection from isomorphism classes of representations in C Q , m to reverse plane partitions whose shape is the minuscule poset corresponding to Q and m. By relating the piecewise-linear promotion action on reverse plane partitions to Auslander–Reiten translation in the derived category, we give a uniform proof that the order of promotion equals the Coxeter number. In type A n , we show that special cases of our bijection include the Robinson–Schensted–Knuth and Hillman–Grassl correspondences. [ABSTRACT FROM AUTHOR]

We provide a quantum Laurent expansion formula for the quantum cluster algebras from unpunctured orbifolds with arbitrary coefficients and quantization. As an application, positivity for such a class of quantum cluster algebras is provided. Assume for technical reasons that the weights of the orbifold points are 1/2. [ABSTRACT FROM AUTHOR]

An exchange matrix B dominates an exchange matrix B' if the signs of corresponding entries weakly agree, with the entry of B always having weakly greater absolute value. When B dominates B', interesting things happen in many cases (but not always): the identity map between the associated mutation-linear structures is often mutation-linear; the mutation fan for B often refines the mutation fan for B'; the scattering (diagram) fan for B often refines the scattering fan for B'; and there is often an injective homomorphism from the principal-coefficients cluster algebra for B' to the principal-coefficients cluster algebra for B, preserving \mathbf {g}-vectors and sending the set of cluster variables for B' (or an analogous larger set) into the set of cluster variables for B (or an analogous larger set). The scope of the description "often" is not the same in all four contexts and is not settled in any of them. In this paper, we prove theorems that provide examples of these dominance phenomena. [ABSTRACT FROM AUTHOR]

A sorting network (also known as a reduced decomposition of the reverse permutation) is a shortest path from 12 \cdots n to n \cdots 21 in the Cayley graph of the symmetric group S_n generated by adjacent transpositions. We prove that in a uniform random n-element sorting network \sigma ^n, all particle trajectories are close to sine curves with high probability. We also find the weak limit of the time-t permutation matrix measures of \sigma ^n. As a corollary of these results, we show that if S_n is embedded into \mathbb {R}^n via the map \tau \mapsto (\tau (1), \tau (2), \dots \tau (n)), then with high probability, the path \sigma ^n is close to a great circle on a particular (n-2)-dimensional sphere in \mathbb {R}^n. These results prove conjectures of Angel, Holroyd, Romik, and Virág. [ABSTRACT FROM AUTHOR]

We introduce the symplectic group Sp 2 (A , σ) over a noncommutative algebra A with an anti-involution σ . We realize several classical Lie groups as Sp 2 over various noncommutative algebras, which provides new insights into their structure theory. We construct several geometric spaces, on which the groups Sp 2 (A , σ) act. We introduce the space of isotropic A-lines, which generalizes the projective line. We describe the action of Sp 2 (A , σ) on isotropic A-lines, generalize the Kashiwara-Maslov index of triples and the cross ratio of quadruples of isotropic A-lines as invariants of this action. When the algebra A is Hermitian or the complexification of a Hermitian algebra, we introduce the symmetric space X Sp 2 (A , σ) , and construct different models of this space. Applying this to classical Hermitian Lie groups of tube type (realized as Sp 2 (A , σ) ) and their complexifications, we obtain different models of the symmetric space as noncommutative generalizations of models of the hyperbolic plane and of the three-dimensional hyperbolic space. We also provide a partial classification of Hermitian algebras in Appendix A. [ABSTRACT FROM AUTHOR]

Assume that the valuation semigroup \Gamma (\lambda) of an arbitrary partial flag variety corresponding to the line bundle \mathcal {L_\lambda } constructed via a full-rank valuation is finitely generated and saturated. We use Ehrhart theory to prove that the associated Newton–Okounkov body — which happens to be a rational, convex polytope — contains exactly one lattice point in its interior if and only if \mathcal {L}_\lambda is the anticanonical line bundle. Furthermore, we use this unique lattice point to construct the dual polytope of the Newton–Okounkov body and prove that this dual is a lattice polytope using a result by Hibi. This leads to an unexpected, necessary and sufficient condition for the Newton–Okounkov body to be reflexive. [ABSTRACT FROM AUTHOR]

We show that the totally nonnegative part of a partial flag variety G/P (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally nonnegative Grassmannian is homeomorphic to a ball, confirming a conjecture of Postnikov. [ABSTRACT FROM AUTHOR]

The support of the tensor product decomposition of integrable irreducible highest weight representations of a symmetrizable Kac-Moody Lie algebra \mathfrak {g} defines a semigroup of triples of weights. Namely, given \lambda in the set P_+ of dominant integral weights, V(\lambda) denotes the irreducible representation of \mathfrak {g} with highest weight \lambda. We are interested in the tensor semigroup \begin{equation*} \Gamma _{\mathbb {N}}(\mathfrak {g})≔\{(\lambda _1,\lambda _2,\mu)\in P_{+}^3\,|\, V(\mu)\subset V(\lambda _1)\otimes V(\lambda _2)\}, \end{equation*} and in the tensor cone \Gamma (\mathfrak {g}) it generates: \begin{equation*} \Gamma (\mathfrak {g})≔\{(\lambda _1,\lambda _2,\mu)\in P_{+,{\mathbb {Q}}}^3\,|\,\exists N\geq 1 \quad V(N\mu)\subset V(N\lambda _1)\otimes V(N\lambda _2)\}. \end{equation*} Here, P_{+,{\mathbb {Q}}} denotes the rational convex cone generated by P_+. In the special case when \mathfrak {g} is a finite-dimensional semisimple Lie algebra, the tensor semigroup is known to be finitely generated and hence the tensor cone to be convex polyhedral. Moreover, the cone \Gamma (\mathfrak {g}) is described in Belkale and Kumar [Invent. Math. 166 (2006), pp. 185–228] by an explicit finite list of inequalities. In general, \Gamma (\mathfrak {g}) is neither polyhedral, nor closed. In this article we describe the closure of \Gamma (\mathfrak {g}) by an explicit countable family of linear inequalities for any untwisted affine Lie algebra, which is the most important class of infinite-dimensional Kac-Moody algebra. This solves a Brown-Kumar's conjecture in this case (see Brown and Kumar [Math. Ann. 360 (2014), pp. 901–936]). The difference between the tensor cone and the tensor semigroup is measured by the saturation factors. Namely, a positive integer d is called a saturation factor , if V(N\lambda _1)\otimes V(N\lambda _2) contains V(N\mu) for some positive integer N then V(d\lambda _1)\otimes V(d\lambda _2) contains V(d\mu), assuming that \mu -\lambda _1-\lambda _2 belongs to the root lattice. For \mathfrak {g}={\mathfrak {sl}}_n, the famous Knutson-Tao theorem asserts that d=1 is a saturation factor (see Knutson and Tao [J. Amer. Math. Soc. 12 (1999), pp. 1055–1090]). More generally, for any simple Lie algebra, explicit saturation factors are known. In the Kac-Moody case, \Gamma _{\mathbb {N}}(\mathfrak {g}) is not necessarily finitely generated and hence the existence of such a factor is unclear a priori. Here, we obtain explicit saturation factors for any affine Kac-Moody Lie algebra. For example, in type \tilde A_n, we prove that any integer d\geq 2 is a saturation factor, generalizing the case \tilde A_1 shown in Brown and Kumar [Math. Ann. 360 (2014), pp. 901–936]. [ABSTRACT FROM AUTHOR]

Let G be a semisimple simply connected complex algebraic group. Let U be the unipotent radical of a Borel subgroup in G. We describe the coordinate rings of U (resp., G/U, G) in terms of two (resp., four, eight) birational charts introduced by Lusztig [ Total positivity in reductive groups , Birkhäuser Boston, Boston, MA, 1994; Bull. Inst. Math. Sin. (N.S.) 14 (2019), pp. 403–459] in connection with the study of total positivity. [ABSTRACT FROM AUTHOR]

The Newell-Littlewood numbers are defined in terms of their celebrated cousins, the Littlewood-Richardson coefficients. Both arise as tensor product multiplicities for a classical Lie group. They are the structure coefficients of the K. Koike-I. Terada basis of the ring of symmetric functions. Recent work of H. Hahn studies them, motivated by R. Langlands' beyond endoscopy proposal; we address her work with a simple characterization of detection of Weyl modules. This motivates further study of the combinatorics of the numbers. We consider analogues of ideas of J. De Loera-T. McAllister, H. Derksen-J. Weyman, S. Fomin–W. Fulton-C.-K. Li–Y.-T. Poon, W. Fulton, R. King-C. Tollu-F. Toumazet, M. Kleber, A. Klyachko, A. Knutson-T. Tao, T. Lam-A. Postnikov-P. Pylyavskyy, K. Mulmuley-H. Narayanan-M. Sohoni, H. Narayanan, A. Okounkov, J. Stembridge, and H. Weyl. [ABSTRACT FROM AUTHOR]

Following a question of Vinberg, a general method to construct monomial bases for finite-dimensional irreducible representations of a reductive Lie algebra g was developed in a series of papers by Feigin, Fourier, and Littelmann. Relying on this method, we construct monomial bases of multiplicity spaces associated with the restriction of the representation to a reductive subalgebra g 0 ⊂ g . As an application, we produce new monomial bases for representations of the symplectic Lie algebra associated with a natural chain of subalgebras. One of our bases is related via a triangular transition matrix to a suitably modified version of the basis constructed earlier by the first author. In type A, our approach shows that the Gelfand–Tsetlin basis and the canonical basis of Lusztig have a common PBW-parameterisation. This implies that the transition matrix between them is triangular. We show also that a similar relationship holds for the Gelfand–Tsetlin and the Littelmann bases in type A. [ABSTRACT FROM AUTHOR]

Let G be a connected semisimple Lie group. There are two natural duality constructions that assign to G: its Langlands dual group G ∨ , and its Poisson–Lie dual group G ∗ , respectively. The main result of this paper is the following relation between these two objects: the integral cone defined by the cluster structure and the Berenstein–Kazhdan potential on the double Bruhat cell G ∨ ; w 0 , e ⊂ G ∨ is isomorphic to the integral Bohr–Sommerfeld cone defined by the Poisson structure on the partial tropicalization of K ∗ ⊂ G ∗ (the Poisson–Lie dual of the compact form K ⊂ G ). By Berenstein and Kazhdan (in: Contemporary mathematics, vol. 433. American Mathematical Society, Providence, pp 13–88, 2007), the first cone parametrizes the canonical bases of irreducible G-modules. The corresponding points in the second cone belong to integral symplectic leaves of the partial tropicalization labeled by the highest weight of the representation. As a by-product of our construction, we show that symplectic volumes of generic symplectic leaves in the partial tropicalization of K ∗ are equal to symplectic volumes of the corresponding coadjoint orbits in Lie (K) ∗ . To achieve these goals, we make use of (Langlands dual) double cluster varieties defined by Fock and Goncharov (Ann Sci Ec Norm Supér (4) 42(6):865–930, 2009). These are pairs of cluster varieties whose seed matrices are transpose to each other. There is a naturally defined isomorphism between their tropicalizations. The isomorphism between the cones described above is a particular instance of such an isomorphism associated to the double Bruhat cells G w 0 , e ⊂ G and G ∨ ; w 0 , e ⊂ G ∨ . [ABSTRACT FROM AUTHOR]

Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel, Kontsevich, and Siebert have given combinatorial constructions of canonical bases of "theta functions" on the coordinate rings of various log Calabi-Yau spaces, including cluster varieties. We prove that the theta bases for cluster varieties are determined by certain descendant log Gromov-Witten invariants of the symplectic leaves of the mirror/Langlands dual cluster variety, as predicted in the Frobenius structure conjecture of Gross-Hacking-Keel. We further show that these Gromov-Witten counts are often given by naive counts of rational curves satisfying certain geometric conditions. As a key new technical tool, we introduce the notion of "contractible" tropical curves when showing that the relevant log curves are torically transverse. [ABSTRACT FROM AUTHOR]

The convolution ring ... was identified with a quantum unipotent cell of the loop group LSL2 in Cautis and Williams [J. Amer. Math. Soc. 32 (2019), pp. 709-778]. We identify the basis formed by the classes of irreducible equivariant perverse coherent sheaves with the dual canonical basis of the quantum unipotent cell. [ABSTRACT FROM AUTHOR]

This paper develops a theory of monoidal categories relative to a braided monoidal category, called augmented monoidal categories. For such categories, balanced bimodules are defined using the formalism of balanced functors. It is shown that there exists a monoidal structure on the relative tensor product of two augmented monoidal categories which is Morita dual to a relative version of the monoidal center. In examples, a category of locally finite weight modules over a quantized enveloping algebra is equivalent to the relative monoidal center of modules over its Borel part. A similar result holds for small quantum groups, without restricting to locally finite weight modules. More generally, for modules over bialgebras inside a braided monoidal category, the relative center is shown to be equivalent to the category of Yetter–Drinfeld modules inside the braided category. If the braided category is given by modules over a quasitriangular Hopf algebra, then the relative center corresponds to modules over a braided version of the Drinfeld double (i.e. the double bosonization in the sense of Majid) which are locally finite for the action of the dual. [ABSTRACT FROM AUTHOR]

We study the relation between quantum affine algebras of type A and Grassmannian cluster algebras. Hernandez and Leclerc described an isomorphism from the Grothendieck ring of a certain subcategory C ℓ of U q (sl n ^) -modules to a quotient of the Grassmannian cluster algebra in which certain frozen variables are set to 1. We explain how this induces an isomorphism between the monoid of dominant monomials, used to parameterize simple modules, and a quotient of the monoid of rectangular semistandard Young tableaux with n rows and with entries in [ n + ℓ + 1 ] . Via the isomorphism, we define an element ch(T) in a Grassmannian cluster algebra for every rectangular tableau T. By results of Kashiwara, Kim, Oh, and Park, and also of Qin, every Grassmannian cluster monomial is of the form ch(T) for some T. Using a formula of Arakawa–Suzuki, we give an explicit expression for ch(T), and also give explicit q-character formulas for finite-dimensional U q (sl n ^) -modules. We give a tableau-theoretic rule for performing mutations in Grassmannian cluster algebras. We suggest how our formulas might be used to study reality and primeness of modules, and compatibility of cluster variables. [ABSTRACT FROM AUTHOR]

We propose a new approach to studying electrical networks interpreting the Ohm law as the operator which solves certain local Yang–Baxter equation. Using this operator and the medial graph of the electrical network we define a vertex integrable statistical model and its boundary partition function. This gives an equivalent description of electrical networks. We show that, in the important case of an electrical network on the standard graph introduced in [Curtis E B et al 1998 Linear Algebr. Appl. 283 115–50], the response matrix of an electrical network, its most important feature, and the boundary partition function of our statistical model can be recovered from each other. Defining the electrical varieties in the usual way we compare them to the theory of the Lusztig varieties developed in [Berenstein A et al 1996 Adv. Math. 122 49–149]. In our picture the former turns out to be a deformation of the later. Our results should be compared to the earlier work started in [Lam T and Pylyavskyy P 2015 Algebr. Number Theory 9 1401–18] on the connection between the Lusztig varieties and the electrical varieties. There the authors introduced a one-parameter family of Lie groups which are deformations of the unipotent group. For the value of the parameter equal to 1 the group in the family acts on the set of response matrices and is related to the symplectic group. Using the data of electrical networks we construct a representation of the group in this family which corresponds to the value of the parameter −1 in the symplectic group and show that our boundary partition functions belong to it. Remarkably this representation has been studied before in the work on six vertex statistical models and the representations of the Temperley–Lieb algebra. [ABSTRACT FROM AUTHOR]

A cluster algebra is unistructural if the set of its cluster variables determines its clusters and seeds. It is conjectured that all cluster algebras are unistructural. In this paper, we show that any cluster algebra arising from a triangulation of a marked surface without punctures is unistructural. Our proof relies on the existence of a positive basis known as the bracelet basis and on the skein relations. We also prove that a cluster algebra defined from a disjoint union of quivers is unistructural if and only if the cluster algebras defined from the connected components of the quiver are unistructural. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and study a large class of g-module algebras that we call factorizable by generalizing the Gauss factorization of square or rectangular matrices. This class includes coordinate algebras of corresponding reductive groups G , their parabolic subgroups, basic affine spaces, and many others. It turns out that products of factorizable algebras are also factorizable and it is easy to create a factorizable algebra out of virtually any g-module algebra. We also have quantum versions of all these constructions in the category of Uq(g)-module algebras. Quite surprisingly, our quantum factorizable algebras are naturally acted on by the quantized enveloping algebra Uq(g*) of the dual Lie bialgebra g* of g⁠. [ABSTRACT FROM AUTHOR]

The goal of this paper is to introduce and study noncommutative Catalan numbers C n which belong to the free Laurent polynomial algebra L n in n generators. Our noncommutative numbers admit interesting (commutative and noncommutative) specializations, one of them related to Garsia–Haiman (q, t)-versions, another—to solving noncommutative quadratic equations. We also establish total positivity of the corresponding (noncommutative) Hankel matrices H n and introduce accompanying noncommutative binomial coefficients. [ABSTRACT FROM AUTHOR]

Given a quantum cluster algebra, we show that its triangular bases defined by Berenstein and Zelevinsky and those defined by the author are the same for the seeds associated with acyclic quivers. This result implies that the Berenstein-Zelevinsky basis contains all of the quantum cluster monomials. We also give an easy proof that the two bases are the same for the seeds associated with bipartite skew-symmetrizable matrices. [ABSTRACT FROM AUTHOR]

We construct an injective algebra homomorphism of the quantum group U q (sl n + 1) into a quantum cluster algebra L n associated to the moduli space of framed P G L n + 1 -local systems on a marked punctured disk. We obtain a description of the coproduct of U q (sl n + 1) in terms of the corresponding quantum cluster algebra associated to the marked twice punctured disk, and express the action of the R-matrix in terms of a mapping class group element corresponding to the half-Dehn twist rotating one puncture about the other. As a consequence, we realize the algebra automorphism of U q (sl n + 1) ⊗ 2 given by conjugation by the R-matrix as an explicit sequence of cluster mutations, and derive a refined factorization of the R-matrix into quantum dilogarithms of cluster monomials. [ABSTRACT FROM AUTHOR]

In this paper, we describe a general setting for dimer models on cylinders over Dynkin diagrams which in type A reduces to the well-studied case of dimer models on a disc. We prove that all Berenstein-Fomin-Zelevinsky quivers for Schubert cells in a symmetric Kac-Moody algebra give rise to dimer models on the cylinder over the corresponding Dynkin diagram. We also give an independent proof of a result of Buan, Iyama, Reiten, and Smith that the corresponding superpotentials are rigid using the dimer model structure of the quivers. [ABSTRACT FROM AUTHOR]

A combinatorial expression for the coefficient of the Schur function sλ in the expansion of the plethysm pn/dd ... sμ is given for all d dividing n for the cases in which n = 2 or λ is rectangular. In these cases, the coefficient n/dd ... sμ, sλ> is shown to count, up to sign, the number of fixed points of an μn, sλ>-element set under the dth power of an order-n cyclic action. If n = 2, the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if λ is rectangular, the action is a certain power of Schützenberger and Shimozono's jeu-de-taquin promotion. This work extends results of Stembridge and Rhoades linking fixed points of the Schützenberger actions to ribbon tableaux enumeration. The conclusion for the case n = 2 is equivalent to the domino tableaux rule of Carré and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function. [ABSTRACT FROM AUTHOR]

We give new combinatorial formulas for decomposition of the tensor product of integrable highest weight modules over the classical Lie algebras of types B,C,D, and the branching decomposition of an integrable highest weight module with respect to a maximal Levi subalgebra of type A. This formula is based on a combinatorial model of classical crystals called spinor model. We show that our formulas extend in a bijective way various stable branching rules for classical groups to arbitrary highest weights, including the Littlewood restriction rules. [ABSTRACT FROM AUTHOR]

The article concerns the existence and uniqueness of quantisations of cluster algebras. We prove that cluster algebras with an initial exchange matrix of full rank admit a quantisation in the sense of Berenstein-Zelevinsky and give an explicit generating set to construct all quantisations. [ABSTRACT FROM AUTHOR]

Cactus group is the fundamental group of the real locus of the Deligne-Mumford moduli space of stable rational curves. This group appears naturally as an analog of the braid group in coboundary monoidal categories. We define an action of the cactus group on the set of Bethe vectors of the Gaudin magnet chain corresponding to arbitrary semisimple Lie algebra g. Cactus group appears in our construction as a subgroup in the Galois group of Bethe Ansatz equations. Following the idea of Pavel Etingof, we conjecture that this action is isomorphic to the action of the cactus group on the tensor product of crystals coming from the general coboundary category formalism. We prove this conjecture in the case g = sl2 (in fact, for this case the conjecture almost immediately follows from the results of Varchenko on asymptotic solutions of the KZ equation and crystal bases). We also present some conjectures generalizing this result to Bethe vectors of shift of argument subalgebras and relating the cactus group with the Berenstein-Kirillov group of piecewise-linear symmetries of the Gelfand-Tsetlin polytope. [ABSTRACT FROM AUTHOR]

The goal of this work is to provide an elementary construction of the canonical basis $$\mathbf B(w)$$ in each quantum Schubert cell $$U_q(w)$$ and to establish its invariance under modified Lusztig's symmetries. To that effect, we obtain a direct characterization of the upper global basis $$\mathbf {B}^{up}$$ in terms of a suitable bilinear form and show that $$\mathbf B(w)$$ is contained in $$\mathbf {B}^{up}$$ and its large part is preserved by modified Lusztig's symmetries. [ABSTRACT FROM AUTHOR]

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 GQ 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 GQ 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(i)n) (i = 1, 2) and finite dimensional graded module categories over the quiver Hecke algebra RAn associated to An through the generalized quantum affine Schur-Weyl duality functor. [ABSTRACT FROM AUTHOR]

We construct families of Newton-Okounkov bodies for the free group character varieties and configuration spaces of any connected reductive group. We use this construction to give a proof that these spaces are Cohen-Macaulay. [ABSTRACT FROM AUTHOR]