1,354,497 research outputs found
Affine Macdonald conjectures and special values of Felder–Varchenko functions
No full textWe refine the statement of the denominator and evaluation conjectures for affine Macdonald polynomials proposed by Etingof–Kirillov Jr. (Duke Math J 78(2):229–256, 1995) and prove the first non-trivial cases of these conjectures. Our results provide a q-deformation of the computation of genus 1 conformal blocks via elliptic Selberg integrals by Felder–Stevens–Varchenko (Math Res Lett 10(5–6):671–684, 2003). They allow us to give precise formulations for the affine Macdonald conjectures in the general case which are consistent with computer computations. Our method applies recent work of the second named author to relate these conjectures in the case of U_q(sl_2) to evaluations of certain theta hypergeometric integrals defined by Felder–Varchenko (Int Math Res Not 21:1037–1055, 2004). We then evaluate the resulting integrals, which may be of independent interest, by well-chosen applications of the elliptic beta integral introduced by Spiridonov (Uspekhi Mat Nauk 56(1(337)):181–182, 2001)
The Varchenko determinant of a Coxeter arrangement
No full textAbstract
The Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula for this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their associated determinants.</jats:p
The Varchenko determinant of a Coxeter arrangement
No full textThe Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula for this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their associated determinants.peer-reviewe
The Mukhin–Varchenko conjecture for type A
No full textAbstract. We present a generalisation of the famous Selberg integral. This confirms the g = An case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for each simple Lie algebra g. Résumé. On présente une généralisation de la bien connue intégrale de Selberg. Cette généralisation vérifie le cas g = An de la conjecture de Mukhin et Varchenko concernant l’existence d’une intégrale de Selberg pour chaque algèbre de Lie simple g
Diagonal form of the Varchenko matrices
No full textAbstract
Varchenko (Adv Math 97(1):110–144, 1993) defined the Varchenko matrix associated with any real hyperplane arrangement and computed its determinant. In this paper, we show that the Varchenko matrix of a hyperplane arrangement has a diagonal form if and only if it is semigeneral, i.e., without degeneracy. In the case of semigeneral arrangement, we present an explicit computation of the diagonal form via combinatorial arguments and matrix operations, thus giving a combinatorial interpretation of the diagonal entries
The Mukhin-Varchenko conjecture for type A
No full textWe present a generalisation of the famous Selberg integral. This confirms the case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for each simple Lie algebra .On présente une généralisation de la bien connue intégrale de Selberg. Cette généralisation vérifie le cas de la conjecture de Mukhin et Varchenko concernant l'existence d'une intégrale de Selberg pour chaque algèbre de Lie simple
Die Varchenko Matrix für Kegel
No full textConsider an arrangement of hyperplanes and assign to each hyperplane a weight. By using this weights Varchenko defines a bilinear form on the vector space freely generated by the regions of the arrangement. We define this bilinear form for cones of the arrangement. Then we show that the determinant of the matrix of the bilinear form restricted to the cone is determined by the combinatorics of the arrangement inside the cone and factors nicely. The resulting theorem induces Varchenko's thereom about the determinant of the matrix of the Varchenko bilinear form.
We consider cones of the braid arrangement which are defined by partially ordered sets. We give a formular for the determinant of a matrices of the bilinear form restricted to one this cones by using properties of the partially ordered set.Betrachte ein Arrangement von Hyperebenen und weise jeder Hyperebene ein Gewicht zu. Unter Verwendung dieser Gewichte definiert Varchenko eine Bilinearform auf den frei von den Gebieten des Arrangements erzeugten Vektorraumes. Wir definieren diese Bilinearform für Kegel des Arrangements und zeigen, dass die Determinante der Matrix der auf den Kegel eingeschränkten Bilinearform durch die Kombinatorik des Arrangements im Kegel bestimmt ist und schön faktorisiert. Das resultierende Theorem umfasst Varchenkos Theorem über die Determinante der Matrix der Varchenko Bilinearform.
Wir betrachten Kegel des Braid Arrangements, die durch partiell geordnete Mengen definiert werden. Wir geben eine Formel für die Determinante der Matrix der Bilinearformen, die durch einen solchen Kegel definiert ist, und verwenden hierfür Eigenschaften der partiell geordneten Menge
Fórmulas de Brion y Lawrence-Varchenko
No full textEl objetivo de este trabajo es presentar una demostración asequible de las fórmulas de Brion y Lawrence-Varchenko. Éstas no solo permiten contar la totalidad de puntos con coordenadas enteras dentro de un politopo reticular de Rd, sino también mostrar de cuales se trata. Separadamente Michel Brion y James Lawrence junto con Alexander
Varchenko descubrieron, de forma simultánea, dichas fórmulas de gran sencillez que utilizan las funciones generadoras racionales de los conos tangentes a cada vértice del politopo, siendo la fórmula de Lawrence-Varchenko aplicable solo cuando estos conos son simpliciales. El trabajo abarca ambas demostraciones de forma detallada, así como ejemplos que facilitan su comprensión.Grado en Matemática
The signed Varchenko determinant for complexes of oriented matroids.
No full textWe generalize the (signed) Varchenko matrix of a hyperplane arrangement to complexes of oriented matroids and show that its determinant has a nice factorization. This extends previous results on hyperplane arrangements and oriented matroids
Twisted de Rham Complex on Line and Singular Vectors in sl₂ˆ Verma Modules
No full textWe consider two complexes. The first complex is the twisted de Rham complex of scalar meromorphic differential forms on the projective line, holomorphic on the complement to a finite set of points. The second complex is the chain complex of the Lie algebra of sl₂-valued algebraic functions on the same complement, with coefficients in a tensor product of contragradient Verma modules over the affine Lie algebra sl₂ˆ. In [Schechtman V., Varchenko A., Mosc. Math. J. 17 (2017), 787-802] a construction of a monomorphism of the first complex to the second was suggested, and it was indicated that under this monomorphism, the existence of singular vectors in the Verma modules (the Malikov-Feigin-Fuchs singular vectors) is reflected in the relations between the cohomology classes of the de Rham complex. In this paper, we prove these results.The authors thank V. Schechtman for useful discussions. The second author was supported in part by NSF grant DMS-1665239
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