Abstract
Each matrix representation π:G → GLn(ϰ) of a finite Group G over a field ϰ induces an action of G on the module Αn over the polynomial algebra \({\rm A} = \kappa [x_1 , \ldots ,x_n ]\) The graded Α-submodule M(Π) of Αn generated by the orbit of \((x_1 , \ldots ,x_n )\) is studied. A decomposition of M(Π) into generic modules is given. Relations between the numerical invariants of Π and those of M(Π), the latter being efficiently computable by Gröbner bases methods, are examined. It is shown that if Π is multiplicity-free, then the dimensions of the irreducible constituents of Π can be read off from the Hilbert series of M(Pi;). It is proved that determinantal relations form Gröbner bases for the syzygies on generic matrices with respect to any lexicographic order. Gröbner bases for generic modules are also constructed, and their Hilbert series are derived. Consequently, the Hilbert series of M(Pi;) is obtained for an arbitrary representation.
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References
M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison Wesley, Reading, MA, 1969.
D. Bernstein and A. Zelevinsky, “Combinatorics of maximal minors,” J. Alg. Comb. 2 (1993), 111–121.
W. Bruns and U. Vetter, Determinantal Rings, Lecture Notes in Math. 1327, Springer-Verlag, Berlin, 1988.
B. Buchberger, “Gröbner bases-An algorithmic method in polynomial ideal theory,” in Multidimensional Systems Theory, N.K. Bose, ed. D. Reidel, 1985.
D.A. Buchsbaum and D. Eisenbud, “Generic free resolutions and a family of generically perfect ideals,” Adv. Math. 18 (1975), 245–301.
C.W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience, New York, NY, 1962.
J.A. Eagon and D.G. Northcott, “A note on the Hilbert functions of certain ideals which are defined by matrices,” Mathematika 9 (1962), 118–126.
D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Lecture Notes 4, Brandeis University, Waltham, MA., Spring, 1989.
H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin, 1969.
G.D. James. The Representation Theory of the Symmetric Group, Lecture Notes in Mathematics 682, Springer-Verlag, Berlin, 1978.
S. Onn, “Geometry, complexity, and combinatorics of permutation polytopes,” J. Combin. Theory Series A 64 (1993), 31–49.
R.P. Stanley, “Hilbert functions of graded algebras,” Adv. Math. 28 (1978), 57–83.
R.P. Stanley, Combinatorics and Commutative Algebra, Birkhäuser, Boston, 1983.
B. Sturmfels, “Gröbner bases and Stanley decompositions of determinantal rings,” Math. Z. 205 (1990), 137–144.
B. Sturmfels and A. Zelevinsky, “Maximal minors and their leading terms,” Adv. Math. 98 (1993), 65–112.
E.B. Vinberg, Linear Representations of Groups, Birkhäuser, Boston, 1989.
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Onn, S. Hilbert Series of Group Representations and Gröbner Bases for Generic Modules. Journal of Algebraic Combinatorics 3, 187–206 (1994). https://doi.org/10.1023/A:1022445607540
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DOI: https://doi.org/10.1023/A:1022445607540