Abstract
We show that certain homological regularity properties of graded connected algebras, such as being AS-Gorenstein or AS-Cohen–Macaulay, can be tested by passing to associated graded rings. In the spirit of noncommutative algebraic geometry, this can be seen as an analogue of the classical result that, in a flat family of varieties over the affine line, regularity properties of the exceptional fiber extend to all fibers. We then show that quantized coordinate rings of flag varieties and Schubert varieties can be filtered so that the associated graded rings are twisted semigroup rings in the sense of [RZ12]. This is a noncommutative version of the result due to Caldero [C02] stating that flag and Schubert varieties degenerate into toric varieties, and implies that quantized coordinate rings of flag and Schubert varieties are AS-Cohen–Macaulay.
Similar content being viewed by others
References
M. Artin, J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), no. 2, 228–287.
A. Berenstein, A. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), no. 1, 77–128.
A. Björner, F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, Vol. 231, Springer, New York, 2005.
M. P. Brodmann, R. Y. Sharp, Local Cohomology: an Algebraic Introduction with Geometric Applications, Cambridge Studies in Advanced Mathematics, Vol. 60, Cambridge University Press, Cambridge, 1998.
W. Bruns, J. Herzog, Cohen–Macaulay Rings, Cambridge Studies in Advanced Mathematics, Vol. 39, Cambridge University Press, 1993.
W. Bruns, J. Gubeladze, Polytopes, Rings, and K-theory, Springer Monographs in Mathematics, Springer, Dordrecht, 2009.
P. Caldero, Toric degenerations of Schubert varieties, Transform. Groups 7 (2002), no. 1, 51–60.
C. De Concini, D. Eisenbud, C. Procesi, Hodge Algebras, Astérisque 91, Société Mathématique de France, Paris, 1982.
X. Fang, G. Fourier, P. Littelmann, On toric degenerations of flag varieties, in: Representation Theory—Current Trends and Perspectives, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2017, pp. 187–232.
X. Fang, G. Fourier, P. Littelmann, Essential bases and toric degenerations arising from birational sequences, Adv. Math. 312 (2017), 107–149.
J. C. Rosales, P. A. García-Sánchez, Finitely Generated Commutative Monoids, Nova Science Publishers, Inc., Commack, NY, 1999.
N. Gonciulea, V. Lakshmibai, Degenerations of flag and Schubert varieties to toric varieties, Transform. Groups 1 (1996), no. 3, 215–248.
T. Hibi, Distributive lattices, affine semigroup rings and algebras with straightening laws, in: Commutative Algebra and Combinatorics, Kyoto, 1985, Adv. Stud. Pure Math., Vol. 11, North-Holland, Amsterdam, 1987, pp. 93–109.
W. V. D. Hodge, Some enumerative results in the theory of forms, Proc. Cam. Phil. Soc. 39 (1943), 22–30.
J. C. Jantzen, Lectures on Quantum Groups, Graduate Studies in Mathematics, Vol. 6, American Mathematical Society, Providence, RI, 1996.
P. Jørgensen, J. J. Zhang, Gourmet’s guide to Gorensteinness, Adv. Math. 151 (2000), no. 2, 313–345.
M. Kashiwara, The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J. 71 (1993), no. 3, 839–858.
M. Kashiwara, On crystal bases, in: Representations of Groups, Banff, AB, 1994, CMS Conf. Proc., Vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 155–197
V. Lakshmibai, N, Reshetikhin, Quantum flag and Schubert schemes, in: Deformation Theory and Quantum Groups with Applications to Mathematical Physics (Amherst, MA, 1990), Contemp. Math. 134, Amer. Math. Soc., Providence, RI, 1992, pp. 145–181.
T. H. Lenagan, L. Rigal, Quantum graded algebras with a straightening law and the AS-Cohen–Macaulay property for quantum determinantal rings and quantum Grassmannians, J. Algebra 301 (2006), no. 2, 670–702.
P. Littelmann, Cones, crystals, and patterns, Transform. Groups 3 (1998), no. 2, 145–179.
G. Lusztig, Introduction to Quantum Groups, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2010.
G. Maury, J. Raynaud, Ordres Maximaux au Sens de K. Asano, Lecture Notes in Mathematics, Vol. 808, Springer, Berlin, 1980.
J. C. McConnell, J. C., Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics, Vol. 30, American Mathematical Society, Providence, RI. 2001.
C. Năstăsescu, F. Van Oystaeyen, Graded and Filtered Rings and Modules, Lecture Notes in Mathematics, Vol. 758, Springer, Berlin, 1979.
C. Năstăsescu, F. Van Oystaeyen, Methods of Graded Rings, Lecture Notes in Mathematics, Vol. 1836, Springer-Verlag, Berlin, 2004.
L. Rigal, P. Zadunaisky, Quantum analogues of Richardson varieties in the grassmannian and their toric degeneration. J. Algebra 372 (2012), 293–317.
L. Rigal, P. Zadunaisky, Twisted semigroup algebras, Alg. Rep. Theory 18 (2015), no. 5, 1155–1186.
Я. С. Сойбельман, О квантовом многообразии флагов, Функц. анализ и его прил. 26 (1992), вып. 3, 90–92. Engl. transl.: Ya. S. Soĭbelman, On the quantum flag manifold, Funct. Anal. Appl. 26 (1992), no. 3, 225–227.
B. Sturmfels, Gröbner Bases and Convex Polytopes, University Lecture Series, Vol. 8, American Mathematical Society, Providence, RI, 1996.
C. A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994.
A. Woo, A. Yong, When is a Schubert variety Gorenstein? Adv. Math 207 (2006), no. 1, 205–220.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
P. Zadunaisky is supported by a CONICET Postdoctoral fellowship
Rights and permissions
About this article
Cite this article
RIGAL, L., ZADUNAISKY, P. QUANTUM TORIC DEGENERATION OF QUANTUM FLAG AND SCHUBERT VARIETIES. Transformation Groups 26, 1113–1143 (2021). https://doi.org/10.1007/s00031-020-09615-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-020-09615-y