Related papers: A new Young wall realization of $B(\lambda)$ and $…

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$. We give a polyhedral realization of the crystal basis for the extremal weight module of extremal weight $\lambda$, where $\lambda$ is an integral weight whose Weyl group…

We give an explicit construction of irreducible modules over Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^{\lambda}$ for finite classical types using a crystal basis theoretic approach. More precisely, for each…

We construct a crystal base of the negative half of a quantum orthosymplectic superalgebra. It can be viewed as a limit of the crystal bases of $q$-deformed irreducible oscillator representations. We also give a combinatorial description of…

In this paper, we introduce the notion of abstract crystals for quantum generalized Kac-Moody algebras and study their fundamental properties. We then prove the crystal embedding theorem and give a characterization of the crystals…

In this paper are introduced two classes of elements in the enveloping algebra $\mathbf{U}(gl(n))$: the \emph{double Young-Capelli bitableaux} $[\ \fbox{$S \ | \ T$}\ ]$ and the \emph{central} \emph{Schur elements}…

The polyhedral realizations for crystal bases of the integrable highest weight modules of $U_q(\mathfrak{g})$ have been introduced in ([T.Nakashima, J. Algebra, vol.219, no. 2, (1999)]), which describe the crystal bases as sets of lattice…

We study a class of algebras B(n,l) associated with integrable models with boundaries. These algebras can be identified with coideal subalgebras in the Yangian for gl(n). We construct an analog of the quantum determinant and show that its…

We generalize the results of [KMST] concerning equivariant quantization by means of Verma modules $M(\lambda)$ for generic weight $\lambda$ to the case of general $\lambda$. We consider the relationship between the Shapovalov form on an…

We define geometric/unipotent crystal structure on unipotent subgroups of semi-simple algebraic groups. We shall show that in $A_n$-case, their ultra-discretizations coincide with crystals obtained by generalizing Young tableaux.

Given any pair of positive integers m and n, we construct a new Hopf algebra, which may be regarded as a degenerate version of the quantum group of gl(m+n). We study its structure and develop a highest weight representation theory. The…

We study a finite-dimensional algebra $\Lambda$ constructed from a Postnikov diagram $D$ in a disk, obtained from the dimer algebra of Baur-King-Marsh by factoring out the ideal generated by the boundary idempotent. Thus $\Lambda$ is…

We give the crystal structure of the Grothendieck group $G_0(R)$ of irreducible modules over the quiver Hecke algebra $R$ constructed in \cite{TW2023}. This leads to the categorification of the crystal $B(\infty)$ of the quantum Borcherds…

The classical Gindikin-Karpelevich formula appears in Langlands' calculation of the constant terms of Eisenstein series on reductive groups and in Macdonald's work on p-adic groups and affine Hecke algebras. The formula has been generalized…

We study the crystal base of the negative part of a quantum group. Two explicit descriptions of the crystal $B(\infty)$ for types $G_2$ are given. The first is given in terms of extended Nakajima monomials and the second realization follows…

We introduce the notion of dual perfect bases and dual perfect graphs. We show that every integrable highest weight module $V_q(\lambda)$ over a quantum generalized Kac-Moody algebra $U_{q}(\mathcal{g})$ has a dual perfect basis and its…

Truncated shifted Yangians are a family of algebras which are natural quantizations of slices in the affine Grassmannian. We study the highest weight representations of these algebras. In particular, we conjecture that the possible highest…

We consider the crystal structure of the level zero extremal weight modules $V(\lambda)$ using the crystal base of the quantum affine algebra constructed by Beck, Chari and Pressley. This approach yields an explicit form for the U^-…

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and set $\lambda: = \Lambda_1 - \Lambda_2$, where $\Lambda_1, \Lambda_2$ are the fundamental weights for $\mathfrak{g}$; note that $\lambda$ is neither dominant nor…

We show that if a module M over a basic classical Lie superalgebra of type type I is simultaneously a Verma module with respect to some Borel \(\mathfrak b_1\) and a dual Verma module with respect to Borel \(\mathfrak b_2\), then M is…

We construct the Fock space representations of classical quantum affine algebras using combinatorics of Young walls. We also show that the crystal graphs of the Fock space representations can be realized as the abstract crystal consisting…