The conjecturally perfect Kirillov-Reshetikhin (KR) crystals are known to be isomorphic as classical crystals to certain Demazure subcrystals of crystal graphs of irreducible highest weight modules over affine algebras. Under some assumptions we show that the classical isomorphism from the Demazure crystal to the KR crystal, sends zero arrows to zero arrows. This implies that the affine crystal structure on these KR crystals is unique.
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This talk will be about walking around in compact Lie groups, in p-adic groups, in buildings, and in analogues of the upper half plane; with LOTS of pictures. The story is somewhat chronological. In the beginning of representation theory, Frobenius and Schur used partitions and 'Young tableaux' to analyze the representations of their favourite groups, the symmetric group and the general linear group. Hermann Weyl had a great insight and explained how to generalize the partitions to all compact Lie groups. In this case the combinatorics is controlled by the "Weyl group", a group generated by reflections that acts on an integral lattice. Generalising the Young tableaux to all compact Lie groups had to wait until 1994, when P. Littelmann introduced his 'path model' (which was, for him, a combinatorial way to write down the 'crystal' for the corresponding quantum group). A thorn in the side of many researchers has been the fact that there are plenty of groups generated by reflections that "look like" Weyl groups but don't seem to have any compact Lie group associated to them. Homotopy theory has produced (about 1995) the p-compact groups, and proved (about 2004) that they are absolutely the right generalisations of compact Lie groups (at least for this theory). They are constructed as (p-completed) classifying spaces of certain discrete groups. Amazingly, in the classical compact Lie groups case, the classifying space construction matches up with the 'path model' and so it turns out that we arrive at the same object from both the algebraic/ combinatorial and the homotopy theory points of view!
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We describe a path model for Chevalley groups, and discuss the connections to Hecke algebras and Mirković-Vilonen cycles in the loop Grassmanian.
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The algebras NCSym_n and Sym_n (n in N₊) are defined to be the S_n-invariants inside Q< A_n > (respectively Q[X_n]), the polynomial functions on a noncommutative alphabet A_n (respectively commutative, X_n) of cardinality n. The abelianization (a_n |--> x_n) realizes Sym_n as a quotient of NCSym_n. Here, we view it as a subspace. We realize Sym_n as the S_n-invariants inside NCSym_n for a second, natural action of the symmetric group on NCSym_n and describe the coinvariants explicitly. Some surprising identities on the ordinary generating function for the Bell numbers appear as an immediate corollary. In case n=\infty, we obtain new information on the (Hopf) algebraic structure of NCSym_n .

Time permitting, we outline similar results for Hivert's r-QSym_n algebras (r,n in N₊ \cup \infty) and their noncommutative analogues. The algebra Sym_n and Gessel's quasisymmetric functions appear at the extremal values of r.
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Triangulations of a convex polygon are known to be counted by the Catalan numbers. A natural generalization of a triangulation is a k-triangulation, which is defined to be a maximal set of diagonals so that no k + 1 of them mutually cross in their interiors. It was proved by Jonsson that k-triangulations are enumerated by certain determinants of Catalan numbers, that are also known to count k-tuples of non-crossing Dyck paths. There are several simple bijections between triangulations of a convex n-gon and Dyck paths. However, no bijective proof of Jonsson's result is known for general k. Here we solve this problem for k = 2, that is, we present a bijection between 2-triangulations of a convex n-gon and pairs (P,Q) of Dyck paths of semilength n-4 so that P never goes below Q. The bijection is obtained by constructing isomorphic generating trees for the sets of 2-triangulations and pairs of non-crossing Dyck paths.
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In joint work with A. Postnikov, we defined a simple combinatorial model for the irreducible representations of complex semisimple Lie algebras, which will be referred to as the alcove path model. This model was also extended to complex symmetrizable Kac-Moody algebras. It can be viewed as a discrete counterpart to the Littelmann path model. While the main features of Littelmann's model were recovered in the alcove path model, the latter has some additional features too, developed in further solo work. The talk will focus on one of the mentioned additional features in the finite case, namely a combinatorial realization of Lusztig's involution on irreducible crystals. This involution exhibits a crystal as a self-dual poset, and corresponds to the action of the longest Weyl group element on the corresponding representation.
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The irreducible representations of the symmetric group S_n are parameterized by partitions of n. One can use the partition, viewed as being built up row by row, to construct the module algebraically, piece by piece.

Over a field of characteristic p, the irreducible representations of S_n are parameterized by the ``p-regular'' partitions.

However, the analogous construction of these modules fails. We give an alternate (algebraic) construction of the modules, motivated by viewing the crystal of the basic representation of $widehat{sl}_p as a limit of tensor products of level 1 perfect crystals. This construction relies on the theorem of Grojnowski relating the crystal of the basic representation to the simple S_n-modules and their behavior under restriction to S_n-1.
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Normalized irreducible characters of the symmetric group S(n) can be understood as zonal spherical functions of the Gelfand pair (S(n)xS(n),Diag S(n)). They form an orthogonal basis in the space of the functions on the group S(n) invariant with respect to conjugations by S(n). We consider a different Gelfand pair connected with the symmetric group, that is an ``unbalanced'' Gelfand pair (S(n)xS(n-1),Diag S(n-1)). Zonal spherical functions of this Gelfand pair form an orthogonal basis in a larger space of functions on S(n), namely in the space of functions invariant with respect to conjugations by S(n-1). We refer to these zonal spherical functions as normalized marked (or generalized) characters of S(n). The main discovery of the present work is that these marked characters can be computed on the same level as the irreducible characters of the symmetric group. We give a Murnaghan- Nakayama type rule, a Frobenius type formula, and an analogue of the determinantal formula for the marked characters of S(n).
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It is well-known that the representation theory of many finite unipotent groups is wild (for example, the group of upper-triangular matrices over a finite field with ones on the diagonal). A supercharacter theory is a courser version of the usual character theory that preserves much of the information while becoming more manageable. For example, in the case of the full upper-triangular unipotent group, the supercharacters are indexed by labeled set partitions. This talk describes a supercharacter theory for a large family of unipotent groups related to incidence algebras of posets, called pattern groups. Joint work with P. Diaconis.
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I will discuss recent joint work with Larsen in which we show that certain specializations of BMW-algebras are isomorphic to the symmetric squares of Temperley-Lieb algebras. This allows us to compute the closed images of the corresponding braid group representations and implies an identity between the Kauffman and Jones polynomials due to Lickorish.
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The canonical basis can be parametrized by N-tuple of integers called Lusztig parameters and by N-tuple of integers called string parameters. I will present a twisted version of these parametrizations obtained by acting the Schutzenberger involution. These parametrizations give an easy way to describe MV-cycles and to explicit the isomorphism of crystal between the canonical basis and MV-cycles.
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In 2003, David Radford introduced a new method to construct simple modules for the Drinfel'd double of certain finite-dimensional graded Hopf algebras. For a Hopf algebra H, he established a correspondence between isomorphism +classes of simple modules for the Drinfel'd double D(H) and group-like elements of D(H). The restricted two-parameter quantum group u_{r,s}(sl_n) is a Drinfel'd double under certain conditions on the parameters r and s. I use Radford's method and the computer algebra system Singular::Plural to construct the simple modules for u_{r,s}(sl_3) for different values of r and s. \end{document}
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