Related papers: $B_2$-crystals: axioms, structure, models
For coherent families of crystals of affine Lie algebras of type B^{(1)}_n, D^{(1)}_n, A^{(2)}_{2n} and D^{(2)}_{n+1} we describe the combinatorial R matrix using column insertion algorithms for B,C,D Young tableaux.
We introduce and investigate bucolic complexes, a common generalization of systolic complexes and of CAT(0) cubical complexes. They are defined as simply connected prism complexes satisfying some local combinatorial conditions. We study…
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…
We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed.…
This paper provides results on the modular representation theory of the supergroup $GL(m|n).$ Working over a field of arbitrary characteristic, we prove that the explicit combinatorics of certain crystal graphs describe the representation…
Let $(A, I)$ be a bounded prism, and $X$ be a smooth $p$-adic formal scheme over $\Spf(A/I)$. We consider the notion of crystals on Bhatt--Scholze's prismatic site $(X/A)_{\prism}$ of $X$ relative to $A$. We prove that if $X$ is proper over…
In \cite[\S1.3]{Br2}, some unitary representations of ${\rm GL}_2(\mathbf{Q}_p)$ on $p$-adic Banach spaces are associated to 2-dimensional irreducible crystalline representations of ${\rm Gal}(\bar{\mathbf{Q}}_p)/\mathbf{Q}_p)$. Some…
We introduce constructible directed complexes, a combinatorial presentation of higher categories inspired by constructible complexes in poset topology. Constructible directed complexes with a greatest element, called atoms, encompass common…
We provide the explicit combinatorial structure of the Kirillov-Reshetikhin crystals B^{r,s} of type D_n(1), B_n(1), and A_{2n-1}(2). This is achieved by constructing the crystal analogue sigma of the automorphism of the D_n(1) (resp.…
We give a new realization of arbitrary level perfect crystals and arbitrary level irreducible highest weight crystals of type $D_n^{(1)}$, in the language of Young walls. The notions of splitting of blocks and slices play crucial roles in…
Work of Grantcharov et al. develops a theory of abstract crystals for the queer Lie superalgebra $\mathfrak{q}_n$. Such $\mathfrak{q}_n$-crystals form a monoidal category in which the connected normal objects have unique highest weight…
We characterise ideals in two-dimensional regular local rings that arise as ideals of maximal minors of indecomposable integrally closed modules of rank two.
Similarly to the theory of crystalline cohomology, we give a local description of a prismatic crystal and its cohomology in terms of a $q$-Higgs module and the associated $q$-Higgs complex on the bounded prismatic envelope of an embedding…
Let X be a smooth p-adic formal scheme. We show that integral crystalline local systems on the generic fiber of X are equivalent to prismatic F-crystals over the analytic locus of the prismatic site of X. As an application, we give a…
We show that the different labelings of the crystal graph for irreducible highest weight $\mathcal{U}\_q (\hat{\mathfrak{sl}}\_e)$-modules lead to different labelings of the simple modules for non semisimple Ariki-Koike algebras by using…
This paper explores alternative statements of the axioms for lattice gluing, focusing on lattices that are modular, locally finite, and have finite covers, but may have infinite height. We give a set of "maximal" axioms that maximize what…
A previous work gave a combinatorial description of the crystal $B(\infty)$, in terms of certain simple Young tableaux referred to as the marginally large tableaux, for finite dimensional simple Lie algebras. Using this result, we present…
Kashiwara and Saito have defined a crystal structure on the set of irreducible components of Lusztig's quiver varieties. This gives a geometric realization of the crystal graph of the lower half of the quantum group associated to a…
We extend our generic rigidity theory for periodic frameworks in the plane to frameworks with a broader class of crystallographic symmetry. Along the way we introduce a new class of combinatorial matroids and associated linear…
We define highest weight categorical actions of sl_2 on highest weight categories and show that basically all known examples of categorical sl_2-actions on highest weight categories (including rational and polynomial representations of…