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DECOMPOSITIONS OF BERNSTEIN–SATO POLYNOMIALS AND SLICES

Abstract

Let G be a linearly reductive group acting on a vector space V, and f a semi-invariant polynomial on V. In this paper we study systematically decompositions of the Bernstein–Sato polynomial of f in parallel with some representation-theoretic properties of the action of G on V. We provide a technique based on a multiplicity one property, that we use to compute the Bernstein–Sato polynomials of several classical invariants in an elementary fashion. Furthermore, we derive a “slice method” which shows that the decomposition of V as a representation of G can induce a decomposition of the Bernstein–Sato polynomial of f into a product of two Bernstein–Sato polynomials – that of an ideal and that of a semi-invariant of smaller degree. Using the slice method, we compute Bernstein–Sato polynomials for a large class of semi-invariants of quivers.

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References

  1. [1]

    R. Aragona, Semi-invariants of symmetric quivers of finite type, Algebr. Represent. Theory 16 (2013), no. 4, 1051–1083.

    MathSciNet  Article  Google Scholar 

  2. [2]

    I. Assem, D. Simson, A. Skowroński, Elements of the Representation Theory of Associative Algebras, Vol. 1, Techniques of Representation Theory, London Mathematical Society Student Texts, Vol. 65, Cambridge University Press, Cambridge, 2006.

  3. [3]

    A. Borel, Linear Algebraic Groups, 2nd ed., Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, New York, 1991.

  4. [4]

    N. Budur, M. Mustaţă, M. Saito, BernsteinSato polynomials for arbitrary varieties, Compos. Math. 142 (2006), no. 3, 779–797.

    MathSciNet  Article  Google Scholar 

  5. [5]

    H. Derksen, J. Weyman, Semi-invariants of quivers and saturation for LittlewoodRichardson coefficients, J. Amer. Math. Soc. 13 (2000), 467–479.

    MathSciNet  Article  Google Scholar 

  6. [6]

    H. Derksen, J. Weyman, Generalized quivers associated to reductive groups, Colloq. Math. 94 (2002), no. 2, 151–173.

    MathSciNet  Article  Google Scholar 

  7. [7]

    H. Derksen, J. Weyman, On the canonical decomposition of quiver representations, Compos. Math. 133 (2002), 245–265.

    MathSciNet  Article  Google Scholar 

  8. [8]

    H. Derksen, J. Weyman, Combinatorics of quiver representations, Ann. Inst. Fourier (Grenoble) 61 (2011), 1061–1131.

    MathSciNet  Article  Google Scholar 

  9. [9]

    A. Gyoja, Theory of prehomogeneous vector spaces without regularity condition, Publ. RIMS 27 (1991), 861–922.

    MathSciNet  Article  Google Scholar 

  10. [10]

    R. Howe, T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991), 565–619.

    MathSciNet  Article  Google Scholar 

  11. [11]

    M. Kashiwara, D-modules and Microlocal Calculus, Translations of Mathematical Monographs, Vol. 217, American Mathematical Society, Providence, RI, 2003.

  12. [12]

    T. Kimura, The b-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces, Nagoya Math. J. 85 (1982), 1–80.

    MathSciNet  Article  Google Scholar 

  13. [13]

    T. Kimura, Introduction to Prehomogeneous Vector Spaces, Translations of Mathematical Monographs, Vol. 215, American Mathematical Society, Providence, RI, 2003.

  14. [14]

    J. M. Landsberg, L. Manivel, Series of Lie groups, Michigan Math. J. 52 (2004), no. 2, 453–479.

    MathSciNet  Article  Google Scholar 

  15. [15]

    A. Lőrincz, C. Raicu, J. Weyman, U. Walther, BernsteinSato polynomials for maximal minors and sub-maximal Pfaffians, Adv. Math. 307 (2017), 224–252.

    MathSciNet  Article  Google Scholar 

  16. [16]

    A. C. Lőrincz, BernsteinSato polynomials for quivers, http://opencommons.uconn.edu/dissertations/1111,Doctoral Dissertations 1111, PhD Thesis, 2016.

  17. [17]

    A. C. Lőrincz, Singularities of zero sets of semi-invariants for quivers, arXiv: 1509.04170v2 (2017), to appear in J. Commut. Algebra.

  18. [18]

    A. C. Lőrincz, The b-functions of semi-invariants of quivers, J. Algebra 482 (2017), 346–363.

    MathSciNet  Article  Google Scholar 

  19. [19]

    A. C. Lőrincz, C. Raicu, J. Weyman, Equivariant \( \mathcal{D} \)-modules on binary cubic forms, Comm. Algebra (2019), DOI:https://doi.org/10.1080/00927872.2018.1492590.

  20. [20]

    G. D. Mostow, Fully reducible subgroups of algebraic groups, Amer. J. Math. 78 (1956), no. 1, 200–221.

    MathSciNet  Article  Google Scholar 

  21. [21]

    V. L. Popov, E. B. Vinberg, Invariant theory, in: Algebraic Geometry IV, Encyclopaedia of Mathematical Sciences, Vol. 55, Springer-Verlag, Berlin, 1994, pp. 123–284.

  22. [22]

    C. Procesi, Lie Groups: An Approach Through Invariants and Representations, Universitext, Springer, New York, 2007.

  23. [23]

    C. Riedtmann, Explicit description of generic representations for quivers of type Anor Dn, J. Algebra 452 (2016), 474–486.

    MathSciNet  Article  Google Scholar 

  24. [24]

    F. Sato, K. Sugiyama, Multiplicity one property and the decomposition of b-functions, Internat. J. Math. 17 (2006), 195–229.

    MathSciNet  Article  Google Scholar 

  25. [25]

    M. Sato, Theory of prehomogeneous vector spaces (algebraic part), Nagoya Math. J. 120 (1990), 1–34.

    MathSciNet  Article  Google Scholar 

  26. [26]

    M. Sato, M. Kashiwara, T. Kimura, T. Oshima, Micro-local analysis of prehomogeneous vector spaces, Invent. Math. 62 (1980), 117–179.

    MathSciNet  Article  Google Scholar 

  27. [27]

    M. Sato, T. Kimura, A classiffication of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1–155.

    MathSciNet  Article  Google Scholar 

  28. [28]

    A. Schofield, Semi-invariants of quivers, J. London Math. Soc. 43 (1991), 385–395.

    MathSciNet  Article  Google Scholar 

  29. [29]

    A. Schofield, M. Van den Bergh, Semi-invariants of quivers for arbitrary dimension vectors, Indag. Math. (N.S.) 12 (2001), 125–138.

  30. [30]

    D. A. Shmelkin, Locally semi-simple representations of quivers, Transform. Groups 12 (2007), 153–173.

    MathSciNet  Article  Google Scholar 

  31. [31]

    K. Sugiyama, b-Functions associated with quivers of type A, Transform. Groups 16 (2011), 1183–1222.

    MathSciNet  Article  Google Scholar 

  32. [32]

    K. Ukai, b-Functions of prehomogeneous vector spaces of DynkinKostant type for exceptional groups, Compos. Math. 135 (2003), 49–101.

    MathSciNet  Article  Google Scholar 

  33. [33]

    A. Wachi, Contravariant forms on generalized Verma modules and b-functions, Hiroshima Math. J. 29 (1999), 193–225.

    MathSciNet  Article  Google Scholar 

  34. [34]

    R. Walters, The BernsteinSato b-function of the space of cyclic pairs, Publ. Res. Inst. Math. Sci. 51 (2015), no. 2, 273–288.

    MathSciNet  Article  Google Scholar 

  35. [35]

    J. Weyman, Cohomology of Vector Bundles and Syzygies, Cambridge Tracts in Mathematics, Vol. 149, Cambridge University Press, Cambridge, 2003.

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Correspondence to ANDRÁS CRISTIAN LŐRINCZ.

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LŐRINCZ, A.C. DECOMPOSITIONS OF BERNSTEIN–SATO POLYNOMIALS AND SLICES. Transformation Groups 25, 577–607 (2020). https://doi.org/10.1007/s00031-019-09526-7

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