no. 3
The decomposition formula for Verlinde Sums
[La formule de décomposition des sommes de Verlinde]
Loizides, Yiannis1 ; Meinrenken, Eckhard2
1 Cornell University Dept. of Mathematics 310 Malott Hall Ithaca, NY 14853 (USA)
2 University of Toronto Dept. of Mathematics 40 St. George St. Toronto, ON, M5S 2E4 (Canada)
Annales de l'Institut Fourier, Tome 72 (2022) no. 3, pp. 1207-1248.

Nous prouvons une formule de décomposition des sommes de Verlinde (sommes rationnelles trigonométriques), comme contrepartie discrète de la formule de décomposition de Boysal–Vergne pour les séries de Bernoulli. Motivés par des applications aux formules à point fixe en géométrie hamiltonienne, nous développons une version à valeur dans les formes différentielles des séries de Bernoulli et des sommes de Verlinde, et nous étendons la formule de décomposition à ce contexte plus général.

We prove a decomposition formula for Verlinde sums (rational trigonometric sums), as a discrete counterpart to the Boysal–Vergne decomposition formula for Bernoulli series. Motivated by applications to fixed point formulas in Hamiltonian geometry, we develop differential form valued version of Bernoulli series and Verlinde sums, and extend the decomposition formula to this wider context.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3511
Classification : 40B99, 53D30
Keywords: Verlinde sums, rational trigonometric sums, Bernoulli series
Mot clés : sommes de Verlinde, sommes rationnelles trigonométriques, séries de Bernoulli

Loizides, Yiannis 1 ; Meinrenken, Eckhard 2

1 Cornell University Dept. of Mathematics 310 Malott Hall Ithaca, NY 14853 (USA)
2 University of Toronto Dept. of Mathematics 40 St. George St. Toronto, ON, M5S 2E4 (Canada)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Loizides, Yiannis; Meinrenken, Eckhard. The decomposition formula for Verlinde Sums. Annales de l'Institut Fourier, Tome 72 (2022) no. 3, pp. 1207-1248. doi : 10.5802/aif.3511. https://aif.centre-mersenne.org/articles/10.5802/aif.3511/

[1] Alekseev, Anton; Meinrenken, Eckhard; Woodward, Christopher The Verlinde formulas as fixed point formulas, J. Symplectic Geom., Volume 1 (2001) no. 1, pp. 1-46 | DOI | MR | Zbl

[2] Alekseev, Anton; Meinrenken, Eckhard; Woodward, Christopher Duistermaat–Heckman measures and moduli spaces of flat bundles over surfaces, Geom. Funct. Anal., Volume 12 (2002) no. 1, pp. 1-31 | DOI | MR | Zbl

[3] Baldoni, M. Welleda; Beck, Matthias; Cochet, Charles; Vergne, Michele Volume computation for polytopes and partition functions for classical root systems, Discrete Comput. Geom., Volume 35 (2006) no. 4, pp. 551-595 | DOI | MR | Zbl

[4] Baldoni, M. Welleda; Boysal, Arzu; Vergne, Michèle Multiple Bernoulli series and volumes of moduli spaces of flat bundles over surfaces, J. Symb. Comput., Volume 68 (2015) no. 2, pp. 27-60 | DOI | MR | Zbl

[5] Boysal, Arzu; Vergne, Michèle Paradan’s wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator, Ann. Inst. Fourier, Volume 59 (2009) no. 5, pp. 1715-1752 | DOI | Numdam | MR | Zbl

[6] Boysal, Arzu; Vergne, Michèle Multiple Bernoulli series, an Euler–MacLaurin formula, and wall crossings, Ann. Inst. Fourier, Volume 62 (2012) no. 2, pp. 821-858 | DOI | Numdam | MR | Zbl

[7] Brion, Michel; Vergne, Michèle Arrangement of hyperplanes, I: Rational functions and Jeffrey–Kirwan residue, Ann. Sci. Éc. Norm. Supér., Volume 32 (1999) no. 5, pp. 715-741 | DOI | Numdam | MR | Zbl

[8] Dahmen, Wolfgang; Micchelli, Charles A. The number of solutions to linear Diophantine equations and multivariate splines, Trans. Am. Math. Soc., Volume 308 (1988) no. 2, pp. 509-532 | DOI | MR | Zbl

[9] Guillemin, Victor; Prato, Elisa Heckman, Kostant and Steinberg formulas for symplectic manifolds, Adv. Math., Volume 82 (1990) no. 2, pp. 160-179 | DOI | MR | Zbl

[10] Guillemin, Victor W.; Lerman, Eugene M.; Sternberg, Shlomo On the Kostant multiplicity formula, J. Geom. Phys., Volume 5 (1988) no. 4, pp. 721-750 | DOI | MR | Zbl

[11] Loizides, Yiannis Norm-square localization for Hamiltonian LG-spaces, J. Geom. Phys., Volume 114 (2017), pp. 420-449 | DOI | MR | Zbl

[12] Meinrenken, Eckhard Twisted K-homology and group-valued moment maps, Int. Math. Res. Not., Volume 2012 (2012) no. 20, pp. 4563-4618 | DOI | MR

[13] Paradan, Paul-Emile Localization of the Riemann–Roch character, J. Funct. Anal., Volume 187 (2001) no. 2, pp. 442-509 | DOI | MR | Zbl

[14] Paradan, Paul-Emile Wall-crossing formulas in Hamiltonian geometry, Geometric Aspects of Analysis and Mechanics (Progress in Mathematics), Volume 292, Birkhäuser, 2011, pp. 295-343 | DOI | MR | Zbl

[15] Szenes, András Verification of Verlinde’s formulas for SU (2), Int. Math. Res. Not., Volume 1991 (1991) no. 7, pp. 93-98 | DOI | MR | Zbl

[16] Szenes, András The combinatorics of the Verlinde formula, Vector bundles in algebraic geometry (Durham, 1993) (London Mathematical Society Lecture Note Series), Volume 208, Cambridge University Press, 1995, pp. 241-253 | DOI | MR | Zbl

[17] Szenes, András Iterated residues and multiple Bernoulli polynomials, Int. Math. Res. Not., Volume 18 (1998), pp. 937-956 | DOI | MR | Zbl

[18] Szenes, András Residue theorem for rational trigonometric sums and Verlinde’s formula, Duke Math. J., Volume 118 (2003) no. 2, pp. 189-227 | MR | Zbl

[19] Szenes, András; Vergne, Michèle Residue formulae for vector partitions and Euler–Maclaurin sums, Adv. Appl. Math., Volume 30 (2003) no. 1-2, pp. 295-342 | DOI | MR | Zbl

[20] Szenes, András; Vergne, Michèle [Q,R]=0 and Kostant partition functions, Enseign. Math., Volume 63 (2017) no. 3-4, pp. 471-516 | MR | Zbl

[21] Thaddeus, Michael Conformal field theory and the cohomology of the moduli space of stable bundles, J. Differ. Geom., Volume 35 (1992) no. 1, pp. 131-149 | MR | Zbl

[22] Vergne, Michèle Multiple Bernoulli series and wall crossing (Slides for AMS meeting in San Francisco, 2010) | Numdam | Zbl

[23] Vergne, Michèle Residue formulae for Verlinde sums, and for number of integral points in convex rational polytopes, European women in mathematics (Malta, 2001), World Scientific, 2003, pp. 225-285 | DOI | Zbl

[24] Vergne, Michèle Poisson summation formula and box splines (2013) (https://arxiv.org/abs/1302.6599)

[25] Vergne, Michèle Formal equivariant A ^-class, splines and multiplicities of the index of transversally elliptic operators, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 80 (2016) no. 5, pp. 157-192 | MR | Zbl

[26] Verlinde, Erik Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys., B, Volume 300 (1988), pp. 360-376 | DOI | Zbl

[27] Witten, Edward On quantum gauge theories in two dimensions, Commun. Math. Phys., Volume 141 (1991) no. 1, pp. 153-209 | DOI | MR | Zbl

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