Nullforms, polarization and tensorpowers

Abstract

Part I: Singular Spaces of the Nullcone: Given a complex reductive group G and a complex representation V , one of the main goals of invariant theory is to describe - in terms of generators and relations - the ring of invariant polynomial functions, denoted by O(V )G. However, for most pairs G and V , finding explicitly all generators of O(V )G is very difficult. An important step in this search is to find homogeneous invariants whose zero set is the nullcone Nv ⊂ V , i.e. the zero set of all homogeneous non-constant invariant functions on V . Such invariants are strongly related to O(V )G as Hilbert proved the following result: If f1, . . . , fr are homogeneous invariants whose zero set is equal to Nv then O(V )G is a finitely generated module over the subalgebra C[f1, . . . , fr]. Given some invariants fi O(V )G as above one can apply the so called polarization process to obtain a set of functions lying in O(V ⊕k)G. Our main interest in this work is to analyze whether the set of functions obtained in this manner defines the nullcone NV !k . Due to an observation of Kraft and Wallach, this is equivalent to the question whether for every linear subspace H ⊂ Nv of dimension at most k there exists a one-parameter subgroup  : C*  G such that limt0 (t) ·H = 0. For example, for G = SL2 and V = Vn, the binary forms of degree n, this amounts to the question whether every subspace H that consists of forms having a root of multiplicity greater than n/2. This is indeed the case, as we will see. Furthermore we settle the question for G = SLn and V = S2(Cn)* (symmetric bilinear forms), V = 2(Cn)* (skew-symmetric bilinear forms) and G = SL3 and V = S3(C3)* (ternary cubics). Part II: Multiplicities in Tensor Monomials: There exist a lot of formulas to decompose a tensor product of representations V ⊕ W into a direct sum of irreducible representations with respect to an algebraic group G. However these formulas usually involve summing over the Weyl-group, which makes explicit calculations often tedious. When considering multiple tensor products, i.e. tensor monomials V1⊕n1 ⊕ V2⊕n2 · · · Vr⊕nr, then, even with the use of descent computers, an explicit decomposition is mostly impossible because of the complexity that arises. For this reason problems involving tensor monomials remain challenging. The starting point of this work was the following question asked by Finkelberg: For which (d1, d2, . . . , dn−1) ∈ Nn−1 does the tensor monomial Cn⊕d1 ⊕ 2Cn⊕d2 ⊕ 3Cn⊕d3 ⊕· · · ⊕ n−1Cn⊕dn−1 , considered as SLn-representation, contain the trivial representation exactly once? We solve this problem and some related generalizations. However, representations occuring with multiplicity one in the decomposition of a tensor monomial V1⊕n1 ⊕ V2⊕n2 · · · Vr⊕nr are rather rare as we prove that multiplicities of subrepresentations of tensor monomials grow exponentially with respect to ∑ ni. More precisely, we prove, that if G is a simple complex group and V1, . . . , Vr and W irreducible non-trivial representations then there is a constant N and a real number  > 1 such that if ∑ ni  N then mult(W, V1⊕n1 ⊕ V2⊕n2 · · · Vr⊕nr)  ∑ni unless it is zero. In its current form, this part is a preprint which evolved from my diploma thesis, where I solved special cases of the two main results Theorem A and Theorem C. Part III: The Hilbert Nullcone on Tuples of Matrices and Bilinear Forms: In this joint work with Jan Draisma we explicitly determine the irreducible components of the nullcone of the representation of G on M!p, where either G = SL(W) x SL(V) and M = Hom(V,W) (linear maps), or G = SL(V) and M is one of the representations S2(V*) (symmetric bilinear forms), 2(V*) (skew bilinear forms), or V * ⊕ V * (arbitrary bilinear forms). Here V and W are vector spaces over an algebraically closed field K of characteristic zero. We also answer the question of when the nullcone in M⊕p is defined by the polarisations of the invariants on M; typically, this is only the case if either dimV or p is small. A fundamental tool in our proofs is the Hilbert-Mumford criterion for nilpotency. This preprint has already been accepted for publication in the Mathematische Zeitschrift. I mainly contributed to the first problem we solved: counting and describing the components of the nullcone of the symmetric bilinear forms. Most other cases evolved from this one, however.