Hilbert Series of Group Representations and Gröbner Bases for Generic Modules
Shmuel Onn
DOI: 10.1023/A:1022445607540
Abstract
Each matrix representation A = k[ x 1 , \frac{1}{4} , x n ] {\rm A} = κ[x_1 , \ldots ,x_n ] The graded ( x 1 , \frac{1}{4} , x n ) (x_1 , \ldots ,x_n ) is studied. A decomposition of M( 
) into generic modules is given. Relations between the numerical invariants of and those of M( ), the latter being efficiently computable by Gröbner bases methods, are examined. It is shown that if is multiplicity-free, then the dimensions of the irreducible constituents of 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.
) into generic modules is given. Relations between the numerical invariants of and those of M( ), the latter being efficiently computable by Gröbner bases methods, are examined. It is shown that if is multiplicity-free, then the dimensions of the irreducible constituents of 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.Pages: 187–206
Keywords: Gröbner basis; linear representation; generic module; computational algebra; finite group; Hilbert series
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References
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3. W. Bruns and U. Vetter, Determinantal Rings, Lecture Notes in Math. 1327, Springer-Verlag, Berlin, 1988.
4. B. Buchberger, "Grobner bases-An algorithmic method in polynomial ideal theory," in Multidimensional Systems Theory, N.K. Bose, ed. D. Reidel, 1985.
5. D.A. Buchsbaum and D. Eisenbud, "Generic free resolutions and a family of generically perfect ideals," Adv. Math. 18 (1975), 245-301.
6. C.W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience, New York, NY, 1962.
7. 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.
8. D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Lecture Notes 4, Brandeis University, Waltham, MA., Spring, 1989.
9. H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin, 1969.
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