My RESEARCH INTEREST are located at the intersection of algebraic geometry, combinatorics, and representation theory:

  • Representation theory of semi-simple (complex) Lie algebras, GL_n and other algebraic groups
  • Combinatorics of (monomial) bases and Standard Monomial theory
  • Computeralgebra and Gröbner bases
  • Degenerations of Lie algebras and homogeneous spaces, Quiver Grassmannians
  • Tropical geometry vs. Representation theory
  • Polyhedral geometry vs. Representation theory
  • Weyl modules, Kirillov-Reshetikhin modules of loop algebras

The methods in use may originate from combinatorics and discrete mathematics, heavy use of mathematical software (OSCAR) as well as geometric arguments. Most research is done and will be done together with collaborators from institutes all over the world. Consequently I am PI in several research initiatives.

The SFB/TRR 195 on Symbolic methods in mathematics and their applications with the projects

and the SFB 1481 on Sparsity and Singular structures with the project

The informal Aachen-Bochum-Cologne Darstellungstheorie group (ABCD seminar) with regular workshops every term, the ARTIG (Algebra and Representation Theory in Germany) network (the next meeting will be in Aachen ARTIG#4).

PUBLICATION PROFILES can be found here:

PAPER

  1. Linear degenerate symplectic flag varieties: symmetric degenerations and PBW locus, with M. Boos, G. Cerulli Irelli, and X. Fang, arXiv, 2024
  2. Dynkin abelianisations of flag varieties, with S.V.Enugandla, X.Fang, and C.Steinert, arXiv, 2024
  3. Computing monomial bases in Lie theory using OSCAR, with X.Fang, L. Göttgens, and B.Wilop, arXiv, 2024
  4. FFLV polytopes are string polytopes, with E.Cleusters and F.Lerner, arXiv, 2024
  5. A minimal Gröbner basis for simple sl_n or sp_n-modules, with L.van Ess, arXiv, 2024
  6. Order and chain polytopes for maximal ranked posets, with I.Ahmad and M.Joswig, arXiv, 2023
  7. Specialization map for quiver Grassmannian, with G. Cerulli Irelli, F. Esposito and X.Fang, arXiv, 2022
  8. Degenerate flag varieties in network coding, with G.Nebe, Adv. Math. Communications, 2023
  9. Gröbnes bases for fusion products, with J.Flake and V.Levandovskyy, Algebras and Representation Theory, 2022
  10. PBW filtration and monomial bases for Demazure modules in types A and C., with G.Balla and K.Kambaso, Bei. Algebra and Geometrie, 2022
  11. Degenerate Schubert varieties in type A, with R. Chirivi and X.Fang, Transformation Groups, 2021
  12. Cones from quantum groups to tropical flag varieties, with X.Fang and M.Reineke, J. Algebraic Combinatorics, 2021
  13. PBW degenerations of Lie superalgebras and their typical representations, with D.Kus, J. of Lie Theory, 2021
  14. Linear degenerations of flag varieties: partial flags, defining equations, and group actions, with G. Cerulli Irelli, X. Fang, E.Feigin, M. Reineke, Math. Zeitschrift, 2020
  15. A continuous family of marked poset polytopes, with X.Fang. J.-P- Litza, and C.Pegel, SIAM J.Discrete Mathematics, 2020
  16. The Minkowski property and reflexivity of marked poset polytopes, with X.Fang and C.Pegel, Electr. J. Combinatorics, 2020
  17. String cone and superpotential combinatorics for flag and Schubert varieties in type A, with L.Bossinger, JCTA Series A, 2019
  18. On truncated Weyl modules, with V.Martins and A.Moura, Communications in Algebra, 2019
  19. Weighted PBW degenerations and tropical flag varieties, X.Fang and I.Makhlin, Comm, Cont, Math, 2019
  20. Toric degenerations of Gr(2,n) and Gr(3,6) via plabic graphs. L. Bossinger, X.Fang, M.Hering, M.Lanini, Ann. Comb, 2018
  21. Symmetries on plabic graphs, with X.Fang, CR Math, 2018
  22. Linear degenerations of flag varieties, with G. Cerulli Irelli, X. Fang, E.Feigin, M. Reineke, Math. Zeitschrift, 2017
  23. Degree cones and monomial bases of Lie algebras and Quantum groups, with T.backhaus and X.Fang, Glasgow Math. J, 2017
  24. Favourable modules: filtrations, polytopes, Newton-Okounkov bodies and flat degenerations, with E.Feigin and P.Littelmann, Transformation Groups, 2017
  25. On toric degenerations of flag varieties, with X.Fang and P.Littelmann, EMS, 2017
  26. Essential bases and toric degenerations arising from birational sequences, with X.Fang and P.Littelmann, Adv. Math. 2017
  27. Marked chain-order polytopes, with X.Fang, European J. of Combinatorics, DOI 10.1016/j.ejc.2016.06.007
  28. PBW type filtration for quantum groups of type A_n, with X.Fang and M.Reineke, Journal of Algebra, Volume 449 (1), pp 321-345, 2016.
  29. Torus fixed points in Schubert varieties and Genocchi numbers, with X.Fang, Seminaire Lotharingien de Combinatoire 75 (2016), Article B75f
  30. PBW-degenerated Demazure modules and Schubert varieties for triangular elements, Journal of Combinatorial Theory, Series A, Volume 139, April 2016, Pages 132-152.
  31. Favourable modules: Filtrations, polytopes, Newton-Okounkov bodies and flat degenerations, with E.Feigin and P.Littelmann, Transformation Groups, DOI 10.1007/s00031-016-9389-2, 2016.
  32. Marked poset polytopes: Minkowski sums, indecomposables, and unimodular equivalence, Journal of Pure and Applied Algebra, 220 (2), pp. 606-620, 2016
  33. Minuscule Schubert varieties: Poset polytopes, PBW-degenerated Demazure modules, and Kogan faces, with R.Biswal, Algebras and Representation Theory, DOI 10.1007/s10468-015-9548-5, 2015
  34. New homogeneous ideals for current algebras: filtrations, fusion products, and the Pieri rules, Moscow Mathematical Journal, Volume 15 (1), 2015 pp. 49-72.
  35. The degree of the Hilbert-Poincare polynomial of PBW-graded modules, with T.Backhaus, L.Bossinger and C.Desczyk, C. R. Math. Acad. Sci. Paris 352 (2014), no. 12, 959-963.
  36. Schur positivity and Kirillov-Reshetikhin modules, with D.Hernandez, SIGMA 10 (2014).
  37. Global Weyl modules for equivariant map algebras, with N.Manning and A.Savage, International Mathematics Research Notices, 2014; doi: 10.1093/imrn/rnt231
  38. Weyl modules and Levi subalgebras, Journal of Lie Theory, 24, (2014), No. 2, 503-527.
  39. Extended partial order and applications to tensor products, Australasian J. of Combinatorics 58(1), 178-196, (2014).
  40. Posets, tensor products and Schur positivity, with V.Chari and D.Sagaki, Algebra and Number Theory 8-4 (2014), 933–961. DOI 10.2140/ant.2014.8.933.
  41. On multigraded generalizations of Kirillov-Reshetikhin modules, with A.Bianchi, V.Chari and A.Moura, Algebras and Representation Theory, 2013, 10.1007/s10468-013-9408-0
  42. PBW filtration over Z and compatible bases for simple modules in type A_n and C_n, with E.Feigin and P.Littelmann, Symmetries, Integrable Systems and Representations, Springer Proceedings in Mathematics and Statistics 40.
  43. Global Weyl modules for the twisted loop algebra, with N.Manning and P.Senesi, Abhandlungen a.d. Math. Sem. d. Univ. Hamburg, Volume 83, 1, p.53-82, 2013
  44. Demazure and Weyl modules: The twisted current case, with D.Kus, Transactions of the AMS, Volume 365, p. 6037-6064 (2013).
  45. Local Weyl modules for equivariant map algebras with free abelian group actions, with T.Khandai, D.Kus and A.Savage, Journal of Algebra, Volume 350, p.386-404, (2012) .
  46. PBW filtration and bases for symplectic Lie algebras, with E.Feigin and P.Littelmann, International Mathematics Research Notices, first published online February 17, (2011).
  47. PBW filtration and bases for irreducible modules in type A_n, with E.Feigin and P.Littelmann, Transform. Groups 16 (2011), no. 1, 71-89.
  48. A categorical approach to Weyl modules, with V.Chari and T.Khandai, Transform. Groups 15, no. 3, 517-549, (2010) .
  49. Perfectness of Kirillov-Reshetikhin modules of non-exceptionel type, with M.Okado and A.Schilling, Quantum affine algebras, extended affine Lie algebras, and their applications, 127–143, Contemp. Math., 506, AMS, (2010).
  50. Kirillov-Reshetikhin modules of non-exceptionel type, with M.Okado and A.Schilling, Advances in Mathematics, Volume 222, p. 1080-1116, (2009).
  51. Weyl modules for twisted loop algebras, with V.Chari and P.Senesi, Journal of Algebra, Volume 319, p.5016-5038, (2008).
  52. Demazure structure inside Kirillov-Reshetikhin crystals, with A.Schilling and M.Shimozono, Journal of Algebra, Volume 309, p. 386-404, (2007).
  53. Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, with P.Littelmann, Advances in Mathematics, Volume 211, p. 566-593, (2007).
  54. Tensor product structure of affine Demazure modules and limit constructions, with P.Littelmann, Nagoya Math. Journal., Vol. 182, p. 171-198, (2006).

THESES

  1. Finite-dimensional modules of generalized current algebras: KR crystals, PBW filtrations and Weyl modules, Habilitation, Januar 2013.
  2. Some results on finite dimensional modules of current and loop algebras, Dissertation, Oktober 2007.
  3. Tensor product factorization of affine Demazure modules, Diplomarbeit, September 2004.