We describe the branching rule from to , where the latter is embedded via its action on binary cubic forms. We obtain both a numerical multiplicity formula, as well as a minimal system of generators for the geometric realization of the rule.
Let k be a field and k[x,y] the polynomial ring in two variables over k. Let D be a higher k-derivation on k[x,y] and D̅ the extension of D on k(x,y). We prove that if the kernel of D is not equal to k, then the kernel of D̅ is equal to the quotient field of the kernel of D.
On décrit l’algèbre des invariants de l’action naturelle du groupe sur les pinceaux de formes quintiques binaires.
We give a description of faces, of all codimensions, for the cones spanned by the set of weights associated to the rings of semi-invariants of quivers. For a triple flag quiver and its faces of codimension 1 this description reduces to the result of Knutson-Tao-Woodward on the facets of the Klyachko cone. We give new applications to Littlewood-Richardson coefficients, including a product formula for LR-coefficients corresponding to triples of partitions lying on a wall of the Klyachko cone. We systematically...
Let denote generic binary forms, and let denote their -th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the . As a consequence, we show that each of the higher transvectants is redundant in the sense that it can be completely recovered from and . This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of -representations, and the...
The ring of projective invariants of ordered points on the projective line is one of the most basic and earliest studied examples in Geometric Invariant Theory. It is a remarkable fact and the point of this paper that, unlike its close relative the ring of invariants of unordered points, this ring can be completely and simply described. In 1894 Kempe found generators for this ring, thereby proving the First Main Theorem for it (in the terminology introduced by Weyl). In this paper we compute...
We give a presentation (in terms of generators and relations) of the ring of multisymmetric functions that holds for any commutative ring , thereby answering a classical question coming from works of F. Junker [J1, J2, J3] in the late nineteen century and then implicitly in H. Weyl book “The classical groups” [W].
Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra over a field of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordinates of . In particular, the two nontrivial coordinates in the Anick automorphism are both wild. We establish a...