Related papers: A crystal theoretic method for finding rigged conf…
The generalized quantum group of type $A$ is an affine analogue of quantum group associated to a general linear Lie superalgebra, which appears in the study of solutions to the tetrahedron equation or the three-dimensional Yang-Baxter…
From a quantum $K$-matrix of the fundamental representation, we construct one for the Kirillov-Reshetikhin module by fusion construction. Using the $\imath$crystal theory by the last author, we also obtain combinatorial $K$-matrices…
The behavior of the Kirillov-Schilling-Shimozono bijection is examined under the similarity map on Kirillov-Reshetikhin crystals. It enables us to define this bijection over $\mathbb{Q}$. Conjectures on the extension to $\mathbb{R}$ is also…
We describe a combinatorial realization of the crystals $B(\infty)$ and $B(\lambda)$ using rigged configurations in all symmetrizable Kac-Moody types up to certain conditions. This includes all simply-laced types and all non-simply-laced…
The rigged configuration realization $RC(\infty)$ of the crystal $B(\infty)$ was originally presented as a certain connected component within a larger crystal. In this work, we make the realization more concrete by identifying the elements…
In an earlier work, the authors developed a rigged configuration model for the crystal $B(\infty)$ (which also descends to a model for irreducible highest weight crystals via a cutting procedure). However, the result obtained was only valid…
A bijection is defined from Littlewood-Richardson tableaux to rigged configurations. It is shown that this map preserves the appropriate statistics, thereby proving a quasi-particle expression for the generalized Kostka polynomials, which…
Regular $A_n$-, $B_n$- and $C_n$-crystals are edge-colored directed graphs, with ordered colors $1,2,...,n$, which are related to representations of quantized algebras $U_q(\mathfrak{sl}_{n+1})$, $U_q(\mathfrak{sp}_{2n})$ and…
To extend rational materials design and discovery into the space of metastable polymorphs, rapid and reliable assessment of their lifetimes is essential. Motivated by the early work of Buerger (1951), here we investigate the routes to…
In this paper, we study a tensor product of perfect Kirillov-Reshetikhin crystals (KR crystals for short) whose levels are not necessarily equal. We show that, by tensoring with a certain highest weight element, such a crystal becomes…
We show that a tensor product of nonexceptional type Kirillov--Reshetikhin (KR) crystals is isomorphic to a direct sum of Demazure crystals; we do this in the mixed level case and without the perfectness assumption, thus generalizing a…
The box ball system is studied in the crystal theory formulation. New conserved quantities and the phase shift of the soliton scattering are obtained by considering the energy function (or $H$-function) in the combinatorial $R$-matrix.
We give a realization of the Kirillov--Reshetikhin crystal $B^{1,s}$ using Nakajima monomials for $\widehat{\mathfrak{sl}}_n$ using the crystal structure given by Kashiwara. We describe the tensor product $\bigotimes_{i=1}^N B^{1,s_i}$ in…
The combinatorial R matrices are obtained for a family {B_l} of crystals for U'_q(C^{(1)}_n) and U'_q(A^{(2)}_{2n-1}), where B_l is the crystal of the irreducible module corresponding to the one-row Young diagram of length l. The…
In [Frieden, arXiv:1706.02844], we constructed a geometric crystal on the variety $\mathbb{X}_{k} := {\rm Gr}(k,n) \times \mathbb{C}^\times$ which tropicalizes to the affine crystal structure on rectangular tableaux with $n-k$ rows. In this…
Rigged configurations are combinatorial objects prominent in the study of solvable lattice models. Marginally large tableaux are semi-standard Young tableaux of special form that give a realization of the crystals ${\cal B}(\infty)$. We…
A new fermionic formula for the unrestricted Kostka polynomials of type $A_{n-1}^{(1)}$ is presented. This formula is different from the one given by Hatayama et al. and is valid for all crystal paths based on Kirillov-Reshetihkin modules,…
Frequent Directions, as a deterministic matrix sketching technique, has been proposed for tackling low-rank approximation problems. This method has a high degree of accuracy and practicality, but experiences a lot of computational cost for…
For approximately solving linear ill-posed problems in Hilbert spaces, we investigate the regularization properties of the aggregation method and the RatCG method. These recent algorithms use previously calculated solutions of Tikhonov…
A Littelmann path model is constructed for crystals pertaining to a not necessarily symmetrizable Borcherds-Cartan matrix. Here one must overcome several combinatorial problems coming from the imaginary simple roots. The main results are an…