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

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 3, page 775-803
- ISSN: 1435-9855

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topWehlau, 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 -

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