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Complement Submodular Information Measures for Balanced and Robust Data Selection A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs Laplacian Spectrum of the Weakly Zero-Divisor Graph of a Finite Commutative Ring An identity for second Eulerian numbers via lattice-point counting $t$-tone edge coloring of graphs Constructing Maximal Bumpless Pipedreams for Double Grothendieck Polynomials Mubayi's Polynomial-Ideal Conjecture and Cover-Ideal Turán Methods Implicit Binarization via Complex Phase Dynamics in Combinatorial Optimization The limits of Schur multipliers in Pólya conversion problems for the $q$-permanent function Universality theorems for generalized splines Framing Triangulations for Arbitrary Integer Flow Polytopes On the Common Generalization of Gentle Algebras and Framed Directed Acyclic Graphs The complexity of frugal digraph homomorphisms Chaotic and periodic behavior of jeu de taquin on infinite Young tableaux Enumerating Pattern Avoiding Parking Functions Incidence toric ideals and three-point functions Unique Winning Opening Move in Three-Row Chomp Strong majority colorings of graphs A Balancing Theorem for Spanning Trees of Rectangular Grid Graphs Spectral radius and edge-disjoint connected factors of graphs New invariants for rank metric codes, with applications to the classification of rank two semifields of order 256 Flexible DP-4-coloring of planar graphs without 4-cycles and intersecting triangles Balanced intersection size distributions in projective planes List Reconstruction Problem with List Size Two Is Dimensionality a Barrier for Retrieval Models? The INIEP: Irreducible and Positive Realizations The number of Pfaffian orientations on punctured polygonally cellulated surfaces Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern Finite-state enumeration of adjacency-constrained 132-avoiding permutations AMDS and quantum AMDS Constacyclic codes of length $4p^ς$ over $\mathbb{F}_{{p}^{m}}$
Motivic factorisation of KZ local systems and deformation...
Prakash Belkale, Najmuddin Fakhruddin, Swarnava Mukhopadhyay · 2023-09-29 · via math.CO updates on arXiv.org

Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$. The KZ connection is a connection on the constant bundle associated to a set of $n$ finite dimensional irreducible representations of $\mathfrak{g}$ and a nonzero $κ\in \mathbb{C}$, over the configuration space of $n$-distinct points on the affine line. Via the work of Schechtman--Varchenko and Looijenga, when $κ$ is a rational number the associated local systems can be seen to be realisations of naturally defined motivic local systems. We prove a basic factorisation for the nearby cycles of these motivic local systems as some of the $n$ points coalesce. This leads to the construction of a family (parametrised by $κ$) of deformations over $\mathbb{Z}[t]$ of the representation ring of $\mathfrak{g}$--we call these enriched representation rings--which allows one to compute the ranks of the Hodge filtration of the associated variations of mixed Hodge structure; in turn, this has applications to both the local and global monodromy of the KZ connection. In the case of $\mathfrak{sl}_n$ we give an explicit algorithm for computing all products in the enriched representation rings, which we use to prove that if $1/κ$ is an integer then the global monodromy is finite and scalar. We also prove a similar factorisation result for motivic local systems associated to conformal blocks in genus $0$; this leads to the construction of a family of deformations of the fusion rings. Computations in these rings have potential applications to finiteness of global monodromy. Several open problems and conjectures are formulated. These include questions about motivic BGG-type resolutions and the relationship between the Hodge filtration and the filtration by conformal blocks at varying levels.