Box Graphs and Singular Fibers

We determine the higher codimension fibers of elliptically fibered Calabi-Yau fourfolds with section by studying the three-dimensional N=2 supersymmetric gauge theory with matter which describes the low energy effective theory of M-theory compactified on the associated Weierstrass model, a singular model of the fourfold. Each phase of the Coulomb branch of this theory corresponds to a particular resolution of the Weierstrass model, and we show that these have a concise description in terms of decorated box graphs based on the representation graph of the matter multiplets, or alternatively by a class of convex paths on said graph. Transitions between phases have a simple interpretation as `flopping' of the path, and in the geometry correspond to actual flop transitions. This description of the phases enables us to enumerate and determine the entire network between them, with various matter representations for all reductive Lie groups. Furthermore, we observe that each network of phases carries the structure of a (quasi-)minuscule representation of a specific Lie algebra. Interpreted from a geometric point of view, this analysis determines the generators of the cone of effective curves as well as the network of flop transitions between crepant resolutions of singular elliptic Calabi-Yau fourfolds. From the box graphs we determine all fiber types in codimensions two and three, and we find new, non-Kodaira, fiber types for E_6, E_7 and E_8.Comment: 107 pages, 44 figures, v2: added case of E7 monodromy-reduced fiber

Loopless Algorithms to Generate Maximum Length Gray Cycles wrt. k-Character Substitution

Given a binary word relation $\tau$ onto $A^*$ and a finite language $X\subseteq A^*$, a $\tau$-Gray cycle over $X$ consists in a permutation $\left(w_{[i]}\right)_{0\le i\le |X|-1}$ of $X$ such that each word $w_{[i]}$ is an image under $\tau$ of the previous word $w_{{[i-1]}}$. We define the complexity measure $\lambda_{A,\tau}(n)$, equal to the largest cardinality of a language $X$ having words of length at most $n$, and s.t. some $\tau$-Gray cycle over $X$ exists. The present paper is concerned with $\tau=\sigma_k$, the so-called $k$-character substitution, s.t. $(u,v)\in\sigma_k$ holds if, and only if, the Hamming distance of $u$ and $v$ is $k$. We present loopless (resp., constant amortized time) algorithms for computing specific maximum length $$\sigma_k$-Gray cycles.Comment: arXiv admin note: text overlap with arXiv:2108.1365

Skew Howe duality and limit shapes of Young diagrams

We consider the skew Howe duality for the action of certain dual pairs of Lie groups $(G_1, G_2)$ on the exterior algebra $\bigwedge(\mathbb{C}^{n} \otimes \mathbb{C}^{k})$ as a probability measure on Young diagrams by the decomposition into the sum of irreducible representations. We prove a combinatorial version of this skew Howe for the pairs $(\mathrm{GL}_{n}, \mathrm{GL}_{k})$, $(\mathrm{SO}_{2n+1}, \mathrm{Pin}_{2k})$, $(\mathrm{Sp}_{2n}, \mathrm{Sp}_{2k})$, and $(\mathrm{Or}_{2n}, \mathrm{SO}_{k})$ using crystal bases, which allows us to interpret the skew Howe duality as a natural consequence of lattice paths on lozenge tilings of certain partial hexagonal domains. The $G_1$-representation multiplicity is given as a determinant formula using the Lindstr\"om-Gessel-Viennot lemma and as a product formula using Dodgson condensation. These admit natural $q$-analogs that we show equals the $q$-dimension of a $G_2$-representation (up to an overall factor of $q$), giving a refined version of the combinatorial skew Howe duality. Using these product formulas (at $q =1$), we take the infinite rank limit and prove the diagrams converge uniformly to the limit shape.Comment: 54 pages, 12 figures, 2 tables; v2 fixed typos in Theorem 4.10, 4.14, shorter proof of Theorem 4.6 (thanks to C. Krattenthaler), proved of Conjecture 4.17 in v

Representation Theory of Quivers and Finite Dimensional Algebras

Methods and results from the representation theory of quivers and finite dimensional algebras have led to many interactions with other areas of mathematics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further develop such interactions and to stimulate progress in the representation theory of algebras