# Nullforms, polarization and tensorpowers

Bürgin, Matthias. Nullforms, polarization and tensorpowers. 2006, Doctoral Thesis, University of Basel, Faculty of Science.

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Official URL: http://edoc.unibas.ch/diss/DissB_7697

## 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.
Advisors: Kraft, Hanspeter Schwarz, Gerald W. 05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Algebra (Kraft) Kraft, Hanspeter Thesis Doctoral Thesis 7697 Complete Link to catalogue 1 English doi: 10.5451/unibas-004102036urn: urn:nbn:ch:bel-bau-diss76974 22 Jan 2018 15:50 13 Feb 2009 16:14

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