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In their study of the equivariant K-theory of the generalized flag varieties $G/P$, where $G$ is a complex semisimple Lie group, and $P$ is a parabolic subgroup of $G$, Lenart and Postnikov introduced a combinatorial tool, called the alcove paths model. It provides a model for the highest weight crystals with dominant integral highest weights, generalizing the model by semistandard Young tableaux. In this paper, we prove a simple and explicit formula describing the crystal isomorphism between the alcove paths model and the Gelfand-Tsetlin patterns model for type $A$.
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We give explicit actions of Drinfeld generators on Gelfand-Tsetlin bases of super Yangian modules associated with skew Young diagrams. In particular, we give another proof that these representations are irreducible. We study irreducible tame $mathrm Y(mathfrak{gl}_{1|1})$-modules and show that a finite-dimensional irreducible $mathrm Y(mathfrak{gl}_{1|1})$-module is tame if and only if it is thin. We also give the analogous statements for quantum affine superalgebra of type A.
We provide a classification and an explicit realization of all irreducible Gelfand-Tsetlin modules of the complex Lie algebra sl(3). The realization of these modules uses regular and derivative Gelfand-Tsetlin tableaux. In particular, we list all simple Gelfand-Tsetlin sl(3)-modules with infinite-dimensional weight spaces. Also, we express all simple Gelfand-Tsetlin sl(3)-modules as subquotionets of localized Gelfand-Tsetlin E_{21}-injective modules.
We introduce the notion of essential support of a simple Gelfand-Tsetlin $mathfrak{gl}_n$-module as an important tool towards understanding the character formula of such module. This support detects the weights in the module having maximal possible Gelfand-Tsetlin multiplicities. Using combinatorial tools we describe the essential supports of the simple socles of the universal tableaux modules. We also prove that every simple Verma module appears as a socle of a universal tableaux module and hence obtain a description of the essential supports of all simple Verma modules. As a consequence, we prove the Strong Futorny-Ovsienko Conjecture on the sharpness of the upper bounds of the Gelfand-Tsetlin multiplicities. In addition we give a very explicit description of the support and essential support of the simple singular Verma module $M(-rho)$
We present some results about connections between Littelmann paths and Brownian paths in the framework of affine Lie algebras. We expect that they will be the first steps on a way which could hopefully lead to a Pitman type theorem for a Brownian motion in an alcove associated to an affine Weyl group.
Pitmans theorem states that if {Bt, t $ge$ 0} is a one-dimensional Brownian motion, then {Bt -- 2 inf s$le$t Bs, t $ge$ 0} is a three dimensional Bessel process, i.e. a Brownian motion conditioned in Doob sense to remain forever positive. In this paper one gives a similar representation for the Brownian motion in an interval. Due to the double barrier, it is much more involved and only asymptotic. This uses the fact that the interval is an alcove of the Affine Lie algebra A 1 1 .
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