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Noncommutative matrix integrals and representation varieties of surface groups in a finite group, Annales de l’Institut Fourier 55
, 2005
"... Abstract. A graphical expansion formula for noncommutative matrix integrals with values in a finitedimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their nonorientable counterpart called Möbius graphs. The contribution of each graph is an invariant of the ..."
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Cited by 6 (2 self)
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Abstract. A graphical expansion formula for noncommutative matrix integrals with values in a finitedimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their nonorientable counterpart called Möbius graphs. The contribution of each graph is an invariant of the topological type of the surface on which the graph is drawn. As an example, we calculate the integral on the group algebra of a finite group. We show that the integral is a generating function of the number of homomorphisms from the fundamental group of an arbitrary closed surface into the finite group. The graphical expansion formula yields a new proof of the classical theorems of Frobenius, Schur and Mednykh on these numbers. The purpose of this paper is to establish Feynman diagram expansion formulas for noncommutative matrix integrals over a finitedimensional real or complex von Neumann algebra. An interesting case is the real or complex group algebra of a finite group. Using the graphical expansion formulas, we give a new proof of the classical formulas for the number
A generating function of the number of homomorphisms from a surface group into a finite group
, 209
"... Abstract. A generating function of the number of homomorphisms from the fundamental group of a compact oriented or nonorientable surface without boundary into a finite group is obtained in terms of an integral over a real group algebra. We calculate the number of homomorphisms using the decompositi ..."
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Cited by 5 (3 self)
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Abstract. A generating function of the number of homomorphisms from the fundamental group of a compact oriented or nonorientable surface without boundary into a finite group is obtained in terms of an integral over a real group algebra. We calculate the number of homomorphisms using the decomposition of the group algebra into irreducible factors. This gives a new proof of the classical formulas of Frobenius, Schur, and Mednykh. Let S be a compact oriented or nonorientable surface without boundary, and χ(S) its Euler characteristic. The subject of our study is a generating function of the number Hom(π1(S), G)  of homomorphisms from the fundamental group of S into a finite group G. We give a generating function in terms of a noncommutative integral Eqn.(2.7) or Eqn.(3.2),
Quantization, Classical and Quantum Field Theory and ThetaFunctions.
, 2002
"... Arnaud Beauville’s survey ”Vector bundles on Curves and Generalized Theta functions: Recent Results and Open Problems ” [Be] appeared 10 years ago. This elegant survey is short (16 pages) but provides a complete introduction to a specific part of algebraic geometry. To repeat his succes now we ..."
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Cited by 4 (0 self)
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Arnaud Beauville’s survey ”Vector bundles on Curves and Generalized Theta functions: Recent Results and Open Problems ” [Be] appeared 10 years ago. This elegant survey is short (16 pages) but provides a complete introduction to a specific part of algebraic geometry. To repeat his succes now we
VOLUME OF REPRESENTATION VARIETIES
, 2002
"... Abstract. We introduce the notion of volume of the representation variety of a finitely presented discrete group in a compact Lie group using the pushforward measure associated to a map defined by a presentation of the discrete group. We show that the volume thus defined is invariant under the Andr ..."
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Abstract. We introduce the notion of volume of the representation variety of a finitely presented discrete group in a compact Lie group using the pushforward measure associated to a map defined by a presentation of the discrete group. We show that the volume thus defined is invariant under the AndrewsCurtis moves of the generators and relators of the discrete group, and moreover, that it is actually independent of the choice of presentation if the difference of the number of generators and the number of relators remains the same. We then calculate the volume of the representation variety of a surface group in an arbitrary compact Lie group using the classical technique of Frobenius and Schur on finite groups. Our formulas recover the results of Witten and Liu on the symplectic volume and the Reidemeister torsion of the moduli space of flat Gconnections on a
CAPACITY OF FULLY CORRELATED MIMO SYSTEM USING CHARACTER EXPANSION OF GROUPS
, 2005
"... It is well known that the use of antenna arrays at both sides of communication link can result in high channel capacities provided that the propagation medium is rich scattering. In most previous works presented on MIMO wireless structures, Rayleigh fading conditions were considered. In this work, t ..."
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It is well known that the use of antenna arrays at both sides of communication link can result in high channel capacities provided that the propagation medium is rich scattering. In most previous works presented on MIMO wireless structures, Rayleigh fading conditions were considered. In this work, the capacity of MIMO systems under fully correlated (i.e., correlations between rows and columns of channel matrix) fading is considered. We use replica method and character expansions to calculate the capacity of correlated MIMO channel in closed form. In our calculations, it is assumed that the receiver has perfect channel state information (CSI) but no such information is available at the transmitter. 1.
Phase transitions and random matrices
, 2000
"... Phase transitions generically occur in random matrix models as the parameters in the joint probability distribution of the random variables are varied. They affect all main features of the theory and the interpretation of statistical models. In this paper a brief review of phase transitions in invar ..."
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Phase transitions generically occur in random matrix models as the parameters in the joint probability distribution of the random variables are varied. They affect all main features of the theory and the interpretation of statistical models. In this paper a brief review of phase transitions in invariant ensembles is provided, with some comments to the singular values decomposition in complex nonhermitian ensembles.