A necessary and sufficient condition for the existence of non-trivial S_n-invariants in the splitting algebra

2022/03/17 by Kevin Schlegel, Schlegel, Kevin
Mathematics · Computer Science · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2203.09342

Abstract

For a monic polynomial f over a commutative, unitary ring A the splitting algebra A_f is the universal A-algebra such that f splits in A_f. The symmetric group acts on the splitting algebra by permuting the roots of f. It is known that if the intersection of the annihilators of the elements 2 and D_f (where D_f depends on f) in A is zero, then the invariants under the group action are exactly equal to A. We show that the converse holds.

Related

• On the invariants of the splitting algebra
• Splitting algebras, Symmetric functions and Galois Theory
• On the splitting ring of a polynomial
• Invariants of (-1)-Skew Polynomial Rings under Permutation Representations
• Partition Algebras and the Invariant Theory of the Symmetric Group