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Automorphisms of the ring of invariants of the binary quintic representation of SL2

Daniel Daigle, Gene Freudenburg

Abstract

Let k^[6] denote a polynomial ring in 6 variables over an algebraically closed field k of characteristic zero and consider the action of SL2(k) on k^[6] induced by the irreducible representation of SL2 of degree 5 (the binary quintic representation). We consider the ring Q = (k^[6])^SL2 of invariant polynomials and show that Aut_k(Q) = u(k), the unit group of k, where Aut_k(Q) is the group of k-algebra automorphisms of Q. Based on this result, we show that the group of SL2-equivariant polynomial automorphisms of k^[6] is isomorphic to u(k).

Automorphisms of the ring of invariants of the binary quintic representation of SL2

Abstract

Let k^[6] denote a polynomial ring in 6 variables over an algebraically closed field k of characteristic zero and consider the action of SL2(k) on k^[6] induced by the irreducible representation of SL2 of degree 5 (the binary quintic representation). We consider the ring Q = (k^[6])^SL2 of invariant polynomials and show that Aut_k(Q) = u(k), the unit group of k, where Aut_k(Q) is the group of k-algebra automorphisms of Q. Based on this result, we show that the group of SL2-equivariant polynomial automorphisms of k^[6] is isomorphic to u(k).
Paper Structure (9 sections, 27 theorems, 54 equations)

This paper contains 9 sections, 27 theorems, 54 equations.

Table of Contents

  1. introduction
  2. Acknowledgment.
  3. preliminaries
  4. Locally nilpotent derivations.
  5. The polynomial ring $k^{[3]}$.
  6. Automorphism Groups.
  7. Further preliminary results.
  8. Proofs of the main theorems
  9. Remarks on The binary sextic

Key Result

Theorem 1.1

If $f\in R_9$ satisfies conditions (C1)-(C3) then $|f|_R\ge 2$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • ...and 35 more