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Where can I learn about the invariant theory associated with actions of cyclic groups (as opposed to symmetric groups)?

E.g., do the functions $x+y+z$, $xy+yz+zx$, and $x^2y+y^2z+z^2x$ generate the ring of all cyclically-symmetric polynomials in $x$, $y$, and $z$?

Web-accessible and free references would be preferred.

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Probably here: ? – Mark Sapir Mar 24 '13 at 12:54
There is an example with the cyclic group of order 5 in B.Sturmfels' "Algorithms in invariant theory", Sect. 2.7. The latter section has a general treatment of finite abelian groups, too. It actually looks as if the number of generators of the ring grows quite fast as n increases, e.g. in the case of n=5 you need 11 generators. Counting/finding generators amounts to dealing with certain integer points in a lattice... B.Sturmfels' "Algorithms in invariant theory" (2nd edition) ISBN 978-3-211-77416-8 Springer 2008, Wien New-York – Dima Pasechnik Mar 24 '13 at 16:06
The ring of polynomials invariant under this 3-element cyclic group $G$ cannot be the ring of polynomials in invariants of degree $1,2,3$ because the product of these degrees is $6 \neq |G|$. In fact it is known that a finite subgroup $G$ of ${\rm GL}_n({\bf R})$ has an invariant ring generated by only $n$ polynomials iff $G$ is generated by reflections (linear transformations conjugate to ${\rm diag}(1,1,\ldots,1,-1)$), and the cyclic groups of order $n>2$ do not satisfy this criterion. – Noam D. Elkies Mar 24 '13 at 16:50
You need one more degree three generator in the three-variable case, for example $xyz$. – Tom Goodwillie Mar 24 '13 at 16:51
The ring of fully symmetric polynomials in three variables is freely generated by $x+y+z$, $xy+xz+yz$, and $xyz$. The element $(x-y)(y-z)(z-x)$ is a square root of a polynomial in those three, and surely this yields a presentation of the ring of cyclically invariant functions. – Tom Goodwillie Mar 24 '13 at 17:13

2 Answers 2

A good reference is Richard Stanley's "Invariants of finite groups and their applications to combinatorics," Bull. Amer. Math. Soc. 1 (1979), 475–511. His example 3.6 is a cyclic group of order 4.

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Consider the action of the cyclic group $G$ of order $n$ acting on an $m$-dimensional vector space $V$. I'll assume you are working over an algebraically closed field $k$ (or at least a field containing the nth roots of unity). Over such a field the elements of the cyclic group are diagonalizable. Thus the group is generated by $g=\diag(\omega^{-a_1},\dots,\omega^{-a_m})$ where $\omega$ is a primitive nth root of unity and the $a_i$ are integers (which may be assumed to lie between 0 and $n-1$.

If $\{x_1,\dots,x_m\}$ is a basis for $V^*$ dual to the diagonal basis for $V$, then $g$ acts via $g\cdot x_i = \omega^{a_i} x_i$. Hence if $t = x_1^{e_1}\cdots x_m^{e_m}$ is a monomial then $g\cdot t = \omega^{a_1e_1 + \cdots a_me_m} t$. Since a function is invariant iff it is a linear combination of invariant monomials, the invariant theory reduces to solving the linear congruence $a_1e_1 + \cdots a_me_m \equiv 0 \pmod{n}$ for $(e_1,\dots,e_m)$. These solutions form a monoid in $N^m$. Finding a set of minimal generators for $k[V]^G$ amounts to finding minimal generators for this monoid. This is not really solved in general but there has been a lot of study, as you might imagine.

I suggest the following references.

Weidong Gao, Alfred Geroldinger, Zero-sum problems in finite abelian groups: A survey, Expo. Math. 24 (2006) 337 – 369

John C. Harris and David L. Wehlau, Non-Negative Integer Linear Congruences, Indagationes Mathematicae {\bf 17} No. 1 (2006) 37-44. arXiv:math/0409489v1

Pingzhi Yuan, On the index of minimal zero-sum sequences over finite cyclic groups, Journal of Combinatorial Theory, Series A 114 (2007) 1545–1551

Finally, I'll mention that things are much much more complicated (equivalent to the invariant theory of SL(2,C)) when the characteristic of $k$ divides $n$.

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