Invariants for the modular cyclic group of prime order via classical invariant theory

Wehlau, David L.. "Invariants for the modular cyclic group of prime order via classical invariant theory." Journal of the European Mathematical Society 015.3 (2013): 775-803. <http://eudml.org/doc/277177>.

@article{Wehlau2013,
abstract = {Let $\mathbb \{F\}$ be any field of characteristic $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,\ldots , V_p$ of $C_p$ defined over $\mathbb \{F\}$. Thus if $V$ is any finite dimensional $C_p$-representation there are non-negative integers $0\le n_1,n_2,\ldots ,n_k \le p-1$ such that $V\cong \oplus ^k_\{i=1\}V_\{n_i+1\}$. It is also well-known there is a unique (up to equivalence) $d+1$ dimensional irreducible complex representation of $SL_2(\mathbb \{C\})$ given by its action on the space $R_d$ of $d$ forms. Here we prove a conjecture, made by R. J. Shank, which reduces the computation of the ring of $C_p$-invariants $\mathbb \{F\}\left[\oplus ^k_\{i=1\}V_\{n_i+1\}\right]^\{C_p\}$ to the computation of the classical ring of invariants (or covariants) $\mathbb \{C\}\left[R_1\oplus (\oplus ^k_\{i=1\}R_\{n_i\})\right]^\{SL_2(\mathbb \{C\})\}$. This shows that the problem of computing modular $C_p$ invariants is equivalent to the problem of computing classical $SL_2(\mathbb \{C\})$ invariants. This allows us to compute for the first time the ring of invariants for many representations of $C_p$. In particular, we easily obtain from this generators for the rings of vector invariants $\mathbb \{F\}\left[mV_2\right]^\{c_p\}, \mathbb \{F\}\left[mV_3\right]^\{C_p\}$ and $\mathbb \{F\}\left[mV_4\right]^\{c_p\}$ for all $m\in \mathbb \{N\}$. This is the first computation of the latter two families of rings of invariants.},
author = {Wehlau, David L.},
journal = {Journal of the European Mathematical Society},
keywords = {modular invariant theory; cyclic group; classical invariant theory; Roberts' isomorphism; modular invariant theory; cyclic group; classical invariant theory; Roberts' isomorphism},
language = {eng},
number = {3},
pages = {775-803},
publisher = {European Mathematical Society Publishing House},
title = {Invariants for the modular cyclic group of prime order via classical invariant theory},
url = {http://eudml.org/doc/277177},
volume = {015},
year = {2013},
}

TY - JOUR
AU - Wehlau, David L.
TI - Invariants for the modular cyclic group of prime order via classical invariant theory
JO - Journal of the European Mathematical Society
PY - 2013
PB - European Mathematical Society Publishing House
VL - 015
IS - 3
SP - 775
EP - 803
AB - Let $\mathbb {F}$ be any field of characteristic $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,\ldots , V_p$ of $C_p$ defined over $\mathbb {F}$. Thus if $V$ is any finite dimensional $C_p$-representation there are non-negative integers $0\le n_1,n_2,\ldots ,n_k \le p-1$ such that $V\cong \oplus ^k_{i=1}V_{n_i+1}$. It is also well-known there is a unique (up to equivalence) $d+1$ dimensional irreducible complex representation of $SL_2(\mathbb {C})$ given by its action on the space $R_d$ of $d$ forms. Here we prove a conjecture, made by R. J. Shank, which reduces the computation of the ring of $C_p$-invariants $\mathbb {F}\left[\oplus ^k_{i=1}V_{n_i+1}\right]^{C_p}$ to the computation of the classical ring of invariants (or covariants) $\mathbb {C}\left[R_1\oplus (\oplus ^k_{i=1}R_{n_i})\right]^{SL_2(\mathbb {C})}$. This shows that the problem of computing modular $C_p$ invariants is equivalent to the problem of computing classical $SL_2(\mathbb {C})$ invariants. This allows us to compute for the first time the ring of invariants for many representations of $C_p$. In particular, we easily obtain from this generators for the rings of vector invariants $\mathbb {F}\left[mV_2\right]^{c_p}, \mathbb {F}\left[mV_3\right]^{C_p}$ and $\mathbb {F}\left[mV_4\right]^{c_p}$ for all $m\in \mathbb {N}$. This is the first computation of the latter two families of rings of invariants.
LA - eng
KW - modular invariant theory; cyclic group; classical invariant theory; Roberts' isomorphism; modular invariant theory; cyclic group; classical invariant theory; Roberts' isomorphism
UR - http://eudml.org/doc/277177
ER -