We previously mapped the type 2 diabetes mellitus-2 locus (T2dm2), which affects fasting insulin levels, to distal chromosome 19 in a leptin-deficient obese F2 intercross derived from C57BL/6 (B6) and BTBR T+ tf/J (BTBR) mice. Introgression of a 7-Mb segment of the B6 chromosome 19 into the BTBR background (strain 1339A) replicated the reduced insulin linked to T2dm2. The 1339A mice have markedly impaired insulin secretion in vivo and disrupted islet morphology. We used subcongenic strains derived from 1339A to localize the T2dm2 quantitative trait locus (QTL) to a 242-kb segment comprising the promoter, first exon and most of the first intron of the Sorcs1 gene. This was the only gene in the 1339A strain for which we detected amino acid substitutions and expression level differences between mice carrying B6 and BTBR alleles of this insert, thereby identifying variation within the Sorcs1 gene as underlying the phenotype associated with the T2dm2 locus. SorCS1 binds platelet-derived growth factor, a growth factor crucial for pericyte recruitment to the microvasculature, and may thus have a role in expanding or maintaining the islet vasculature. Our identification of the Sorcs1 gene provides insight into the pathway underlying the pathophysiology of obesity-induced type 2 diabetes mellitus.

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Abstract. Motivated by the counting of BPS states in string theory with orientifolds, we study moduli spaces of self-dual representations of a quiver with contravariant involution. We develop Hall module techniques to compute the number of points over finite fields of moduli stacks of semistable self-dual representations. Wall-crossing formulas relating these counts for different choices of stability parameters recover the wall-crossing of orientifold BPS/DonaldsonThomas invariants predicted in the physics literature. In finite type examples the wall-crossing formulas can be reformulated in terms of identities for quantum dilogarithms acting in representations of quantum tori.

We construct from a finitary exact category with duality A a module over its Hall algebra, called the Hall module, encoding the first order self-dual extension structure of A. We study in detail Hall modules arising from the representation theory of a quiver with involution. In this case we show that the Hall module is naturally a module over the specialized reduced σ-analogue of the quantum Kac-Moody algebra attached to the quiver. For finite type quivers, we explicitly determine the decomposition of the Hall module into irreducible highest weight modules.Date: November 9, 2018. 2010 Mathematics Subject Classification. Primary: 16G20 ; Secondary 17B37. Key words and phrases. Representations of quivers, Hall algebras, quantum groups. 1 2 M. B. YOUNGinterest. In Theorem 2.9, we prove an identity relating E, the Euler form and the stacky number of self-dual extensions in the case that A is hereditary. The proof develops some basic self-dual homological algebra and uses the combinatorics of self-dual analogues of Grothendieck's extensions panachées [12], [4].In Section 3 we study Hall modules arising from the representation theory of a quiver with contravariant involution (Q, σ). From the involution and a choice of signs we define a duality structure on Rep Fq (Q), with q odd. For particular signs, the self-dual objects coincide with the orthogonal and symplectic representations of Derksen and Weyman [7]. The module and comodule structures are incompatible in that M Q is not a Hopf module, even in a twisted sense. In Theorem 3.5, we instead show that the action and coaction of the simple representationsThe proof is combinatorial in nature and involves counting configurations of pairs of self-dual exact sequences, in the spirit of Green's proof of the bialgebra structure of the Hall algebra [11]. We describe in Theorem 3.10 the decomposition of M Q into irreducible highest weight B σ (g Q )-modules. The generators are cuspidal elements of M Q , i.e. elements that are annihilated by the coaction of each [S i ]. The proof relies on a canonically defined non-degenerate bilinear form on the Hall module and a characterization of irreducible highest weight modules due to Enomoto and Kashiwara [9].In Section 4 we restrict attention to finite type quivers. Unlike ordinary quiver representations, self-dual representations in general have non-trivial F q /F q -forms. We extend results of [7] (over algebraically closed fields) to explicitly describe all such forms and classify the indecomposable self-dual F q -representations. The classification is summarized in Theorem 4.2 where a partial root theoretic interpretation of the indecomposables is given. The main application of this result is to the explicit decomposition of Hall modules of finite type quivers into irreducible highest weight B σ (g Q )-modules; see Theorems 4.4 and 4.6. The generators are written as alternating sums of the F q /F q -forms of self-dual indecomposables.In [8], Enomoto proved a result related to Theorems 3.5 and 3.10, showing that induction and restr...

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