Vector invariants of permutation groups in characteristic zero
Date
2023-12-21
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Source Title
International Journal of Mathematics
Print ISSN
0129-167X
Electronic ISSN
1793-6519
Publisher
World Scientific Publishing Co. Pte. Ltd.
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Pages
2350111-1 - 2350111-8
Language
English
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Abstract
We consider a finite permutation group acting naturally on a vector space V over a field k. A well-known theorem of G¨obel asserts that the corresponding ring of invariants k[V ] G is generated by the invariants of degree at most dim V 2 ´ . In this paper, we show that if the characteristic of k is zero, then the top degree of vector coinvariants k[V m]G is also bounded above by
dim V 2 ´ , which implies the degree bound `dim V 2 ´ + 1 for the ring of vector invariants k[V m] G. So, G¨obel’s bound almost holds for vector invariants in characteristic zero as well.