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Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

   
 
 

 

Invariants of the Cox rings of low-complexity double flag varieties for classical groups


Author: E. V. Ponomareva
Translated by: V. E. Nazaikinskii
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 76 (2015), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2015, 71-133
MSC (2010): Primary 14L35; Secondary 14M17
DOI: https://doi.org/10.1090/mosc/244
Published electronically: November 17, 2015
MathSciNet review: 3467261
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Abstract: We find the algebras of unipotent invariants of Cox rings for all double flag varieties of complexity 0 and $ 1$ for the classical groups; namely, we obtain presentations of these algebras. It is well known that such an algebra is simple in the case of complexity 0. We show that, in the case of complexity $ 1$, the algebra in question is either a free algebra or a hypersurface. Knowing the structure of this algebra permits one to effectively decompose tensor products of irreducible representations into direct sums of irreducible representations.


References [Enhancements On Off] (What's this?)

  • 1. É. B. Vinberg and A. L. Onishchik, A seminar on Lie groups and algebraic groups, Nauka, Moscow, 1988. (Russian) MR 1090326 (92i:22014)
  • 2. J. Hausen, Three lectures on Cox rings, LMS Lecture Note Series 405 (2013), 3-60. MR 3077165
  • 3. P. Littelmann, On spherical double cones, J. Algebra 166 (1994), no. 1, 142-157. MR 1276821 (95c:14066)
  • 4. D. I. Panyushev, Complexity and rank of actions in invariant theory, J. Math. Sci. 95 (1999), no. 1, 1925-1985. MR 1708594 (2000h:14041)
  • 5. E. V. Ponomareva, Classification of double flag varieties of complexity 0 and $ 1$, Izv. Ross. Akad. Nauk Ser. Mat. 77 (2013), no. 5, 155-178; English transl., Izv. Math. 77 (2013), no. 5, 998-1020. MR 3137198
  • 6. J. Stembridge, Multiplicity-free products and restrictions of Weyl characters, Represent. Theory. 7 (2003), 404-439. MR 2017064 (2004j:17013)
  • 7. D. A. Timashev, Homogeneous spaces and equivariant embeddings, Encyclopedia of Math. Sci., vol. 138, Springer, 2011. MR 2797018 (2012e:14100)

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Additional Information

E. V. Ponomareva
Affiliation: Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
Email: lizaveta@yandex.ru

DOI: https://doi.org/10.1090/mosc/244
Keywords: Double flag variety, Cox ring, complexity, linear representation, tensor product of representations, branching problem.
Published electronically: November 17, 2015
Article copyright: © Copyright 2015 E. V. Ponomareva

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