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Register a new userAdams operations on the Green ring of a cyclic group of prime-power order
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We consider the Green ring $R_{KC}$ for a cyclic $p$-group $C$ over a field $K$ of prime characteristic $p$ and determine the Adams operations $psi^n$ in the case where $n$ is not divisible by $p$. This gives information on the decomposition into indecomposables of exterior powers and symmetric powers of $KC$-modules.
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The Adams operations $psi_Lambda^n$ and $psi_S^n$ on the Green ring of a group $G$ over a field $K$ provide a framework for the study of the exterior powers and symmetric powers of $KG$-modules. When $G$ is finite and $K$ has prime characteristic $p$ we show that $psi_Lambda^n$ and $psi_S^n$ are periodic in $n$ if and only if the Sylow $p$-subgroups of $G$ are cyclic. In the case where $G$ is a cyclic $p$-group we find the minimum periods and use recent work of Symonds to express $psi_S^n$ in terms of $psi_Lambda^n$.
In this paper, we study the vector invariants, ${bf{F}}[m V_2]^{C_p}$, of the 2-dimensional indecomposable representation $V_2$ of the cylic group, $C_p$, of order $p$ over a field ${bf{F}}$ of characteristic $p$. This ring of invariants was first studied by David Richman cite{richman} who showed that this ring required a generator of degree $m(p-1)$, thus demonstrating that the result of Noether in characteristic 0 (that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group) does not extend to the modular case. He also conjectured that a certain set of invariants was a generating set with a proof in the case $p=2$. This conjecture was proved by Campbell and Hughes in cite{campbell-hughes}. Later, Shank and Wehlau in cite{cmipg} determined which elements in Richmans generating set were redundant thereby producing a minimal generating set. We give a new proof of the result of Campbell and Hughes, Shank and Wehlau giving a minimal algebra generating set for the ring of invariants ${bf{F}}[m V_2]^{C_p}$. In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis for ${bf{F}}[m V_2]^{C_p}$. Further, our techniques also serve to give an explicit decomposition of ${bf{F}}[m V_2]$ into a direct sum of indecomposable $C_p$-modules. Finally, noting that our representation of $C_p$ on $V_2$ is as the $p$-Sylow subgroup of $SL_2({bf F}_p)$, we are able to determine a generating set for the ring of invariants of ${bf{F}}[m V_2]^{SL_2({bf F}_p)}$.
We prove that exterior powers of (skew-)symmetric bundles induce a $lambda$-ring structure on the ring $GW^0(X) oplus GW^2(X)$, when $X$ is a scheme where $2$ is invertible. Using this structure, we define stable Adams operations on Hermitian $K$-theory. As a byproduct of our methods, we also compute the ternary laws associated to Hermitian $K$-theory.
Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(x_1g)cdotldotscdot(x_lg)$ where $gin G$ and $x_1, ldots, x_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(x_1+cdots+x_l)/ord(g)$ over all possible $gin G$ such that $langle g rangle =G$. Recently the second and the third authors determined the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime order where $S=g^2(x_2g)(x_3g)(x_4g)$. In this paper, we determine the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime power order. It is shown that if $G=langle grangle$ is a cyclic group of prime power order $n=p^mu$ with $p geq 7$ and $mugeq 2$, and $S=(x_1g)(x_2g)(x_2g)(x_3g)(x_4g)$ with $x_1=x_2$ is a minimal zero-sum sequence with $gcd(n,x_1,x_2,x_3,x_4,x_5)=1$, then $ind(S)=2$ if and only if $S=(mg)(mg)(mfrac{n-1}{2}g)(mfrac{n+3}{2}g)(m(n-3)g)$ where $m$ is a positive integer such that $gcd(m,n)=1$.
As the size of data storing arrays of disks grows, it becomes vital to protect data against double disk failures. A popular method of protection is via the Reed-Solomon (RS) code with two parity words. In the present paper we construct alternative examples of linear block codes protecting against two erasures. Our construction is based on an abstract notion of cone. Concrete cones are constructed via matrix representations of cyclic groups of prime order. In particular, this construction produces EVENODD code. Interesting conditions on the prime number arise in our analysis of these codes. At the end, we analyse an assembly implementation of the corresponding system on a general purpose processor and compare its write and recovery speed with the standard DP-RAID system.
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