294
M. A. Walton
and (13) character formulas do, as well as the Brion formula (7) for a weight
polytope. Similarities in the formulas given here may indicate that we are on the
right track.
Acknowledgments I thank Jørgen Rasmussen for collaboration and Chad Povey for 3D-printing
rank-3 weight polytopes. This research was supported by a Discovery Grant from the Natural
Sciences and Engineering Research Council of Canada (NSERC).
References
1. H.H. Andersen, Schubert varieties and Demazure’s character formula. Invent. Math. 79, 611–
618 (1985)
2. J.-P. Antoine, D. Speiser, Characters of irreducible representations of the simple groups. I.
General theory. J. Math. Phys. 5, 1226–1234 (1964); Characters of irreducible representations
of the simple groups. II. Application to classical groups. J. Math. Phys. 5, 1560–1572 (1964)
3. M. Brion, Points entiers dans les polyèdres convexes. Ann. Scient. Éc. Norm. Sup., 4e série, t.
21, 653–663 (1988)
4. M. Brion, Polyèdres et réseaux, Enseign. Math. 38(1–2), 71–88 (1992)
5. M. Demazure, Désingularisation des variétés de Schubert généralisées. Ann. scient. Éc. Norm.
sup., t. 6, Sect. 2, 53–88 (1974); Une nouvelle formule des caractères. Bull. Sci. Math. 98(3),
163–172 (1974)
6. G. Dhillon, A. Khare, Characters of highest weight modules and integrability (2016).
arXiv:1606.09640; The Weyl-Kac weight formula. Séminaire Lotharingien de Combinatoire
78B (2017), Proceedings of the 29th Conference on Formal Power Series and Algebraic
Combinatorics (London), Article #77 (2018). arXiv:1802.06974
7. A. Joseph, On the Demazure character formula. Ann. sclent. Éc. Norm. Sup., 4 e série, t. 18,
389 -419 (1985)
8. S. Kass, A recursive formula for characters of simple Lie algebras. J. Alg. 137, 126 (1991)
9. A. Kuniba, K. Misra, M. Okado, T. Takagi, J. Uchiyama, Characters of Demazure modules and
solvable lattice models. Nucl. Phys. B 510, 555–576 (1998)
10. P. Littelmann, A generalization of the Littlewood-Richardson rule. J. Alg. 130, 328–368 (1990)
11. A. Postnikov, Permutohedra, associahedra, and beyond. Int. Math. Res. Not. 6, 1026–1106
(2009)
12. J. Rasmussen, Layer structure of irreducible Lie algebra modules (2018). Preprint
arXiv:1803.06592
13. W. Schutzer, A new character formula for Lie algebras and Lie groups. J. Lie Theory 22(3),
817–838 (2012)
14. M.A. Walton, Demazure characters and WZW fusion rules. J. Math. Phys. 39, 665–681 (1998)
15. M.A. Walton, Polytope sums and Lie characters, in Symmetry in Physics. CRM Proceedings
& Lecture Notes, vol. 34 (American Mathematical Society, Providence, 2004), pp. 203–214;
Proceedings of a CRM Workshop Held in Memory of Robert T. Sharp (2002), pp. 12–14
16. M.A. Walton, Polytope expansion of Lie characters and applications. J. Math. Phys. 54, 121701
(2013)
M. A. Walton
and (13) character formulas do, as well as the Brion formula (7) for a weight
polytope. Similarities in the formulas given here may indicate that we are on the
right track.
Acknowledgments I thank Jørgen Rasmussen for collaboration and Chad Povey for 3D-printing
rank-3 weight polytopes. This research was supported by a Discovery Grant from the Natural
Sciences and Engineering Research Council of Canada (NSERC).
References
1. H.H. Andersen, Schubert varieties and Demazure’s character formula. Invent. Math. 79, 611–
618 (1985)
2. J.-P. Antoine, D. Speiser, Characters of irreducible representations of the simple groups. I.
General theory. J. Math. Phys. 5, 1226–1234 (1964); Characters of irreducible representations
of the simple groups. II. Application to classical groups. J. Math. Phys. 5, 1560–1572 (1964)
3. M. Brion, Points entiers dans les polyèdres convexes. Ann. Scient. Éc. Norm. Sup., 4e série, t.
21, 653–663 (1988)
4. M. Brion, Polyèdres et réseaux, Enseign. Math. 38(1–2), 71–88 (1992)
5. M. Demazure, Désingularisation des variétés de Schubert généralisées. Ann. scient. Éc. Norm.
sup., t. 6, Sect. 2, 53–88 (1974); Une nouvelle formule des caractères. Bull. Sci. Math. 98(3),
163–172 (1974)
6. G. Dhillon, A. Khare, Characters of highest weight modules and integrability (2016).
arXiv:1606.09640; The Weyl-Kac weight formula. Séminaire Lotharingien de Combinatoire
78B (2017), Proceedings of the 29th Conference on Formal Power Series and Algebraic
Combinatorics (London), Article #77 (2018). arXiv:1802.06974
7. A. Joseph, On the Demazure character formula. Ann. sclent. Éc. Norm. Sup., 4 e série, t. 18,
389 -419 (1985)
8. S. Kass, A recursive formula for characters of simple Lie algebras. J. Alg. 137, 126 (1991)
9. A. Kuniba, K. Misra, M. Okado, T. Takagi, J. Uchiyama, Characters of Demazure modules and
solvable lattice models. Nucl. Phys. B 510, 555–576 (1998)
10. P. Littelmann, A generalization of the Littlewood-Richardson rule. J. Alg. 130, 328–368 (1990)
11. A. Postnikov, Permutohedra, associahedra, and beyond. Int. Math. Res. Not. 6, 1026–1106
(2009)
12. J. Rasmussen, Layer structure of irreducible Lie algebra modules (2018). Preprint
arXiv:1803.06592
13. W. Schutzer, A new character formula for Lie algebras and Lie groups. J. Lie Theory 22(3),
817–838 (2012)
14. M.A. Walton, Demazure characters and WZW fusion rules. J. Math. Phys. 39, 665–681 (1998)
15. M.A. Walton, Polytope sums and Lie characters, in Symmetry in Physics. CRM Proceedings
& Lecture Notes, vol. 34 (American Mathematical Society, Providence, 2004), pp. 203–214;
Proceedings of a CRM Workshop Held in Memory of Robert T. Sharp (2002), pp. 12–14
16. M.A. Walton, Polytope expansion of Lie characters and applications. J. Math. Phys. 54, 121701
(2013)