16 research outputs found

### Disentangling the Seesaw in the Left-Right Model -- An Algorithm for the General Case

Senjanovic and Tello have analyzed how one could determine the neutrino Dirac
mass matrix in the minimal left-right model, assuming that the mass matrices
for the light and heavy neutrinos could be taken as inputs. They have provided
an analytical solution for the Dirac mass matrix in the case that the
left-right symmetry is implemented via a generalized parity symmetry and that
this symmetry remains unbroken in the Dirac Yukawa sector. We extend the work
of Senjanovic and Tello to the case in which the generalized parity symmetry is
broken in the Dirac Yukawa sector. In this case the elegant method outlined by
Senjanovic and Tello breaks down and we need to adopt a numerical approach.
Several iterative approaches are described; these are found to work in some
cases but to be highly unstable in others. A stable, prescriptive numerical
algorithm is described that works in all but a vanishingly small number of
cases. We apply this algorithm to numerical data sets that are consistent with
current experimental constraints on neutrino masses and mixings. We also
provide some additional context and supporting explanations for the case in
which the parity symmetry is unbroken.Comment: 34 pages, 4 figures; published versio

### Vertices in multiplicative eigenvalue problem for arbitrary groups

We determine, in an inductive framework, the vertices of the polytope
$P(s,K)$ controlling the conjugacy classes of elements which product to one in
the maximal compact subgroup $K$ of a simple complex algebraic group $G$. This
extends earlier work of the authors in related contexts. One feature of this
work is the use of Kontsevich compactifications of the moduli of $P$-bundles
(replacing the use of quot schemes in type A) which are related to
semi-infinite geometry. We also obtain a quantum generalization of Fulton's
conjecture valid for all $G$.Comment: 58 page

### Extremal rays of the embedded subgroup saturation cone

We examine the extremal rays of the cone of dominant weights $(\mu,
\widehat\mu)$ for groups $G\subseteq \widehat G$ for which there exists $N
\gg0$ such that $\left(V(N\mu)\otimes V(N\widehat \mu)\right)^G\ne (0).$ We
exhibit formulas for a class of rays ("type I") on any regular face of the
cone. These rays are identified thanks to a generalization of Fulton's
conjecture, which we prove along the way. We verify that the remaining rays
("type II") on the face are the images of extremal rays for a smaller cone
under a certain map, whose formula is given. A procedure is given for finding
the rays of the cone not on any regular face. This is a generalization of the
work of Belkale and Kiers on extremal rays for the saturated tensor cone; the
specialization is given by $\widehat G = G\times G$ with the diagonal embedding
of $G$. We include several examples to illustrate the formulas.Comment: 50 pages, substantial edits to accommodate $G$ reductive instead of
semisimple, new examples. To appear in Annales de l'Institut Fourie

### Geometric Invariant Theory and Applications to Tensor Products and the Saturation Conjecture

Irreducible representations of a connected, semisimple, complex Lie group are parametrized by dominant, integral weights. If one such Lie group is embedded inside another, it is natural to ask when the tensor product of an irreducible representation of the smaller group with one of the larger group possesses a nontrivial space of invariants (this is equivalent to the representation theory branching problem). Geometric invariant theory is well-equipped to answer the slightly weaker question of existence of asymptotic invariants. After extensive work by several mathematicians, it is known that the solution set to this question is defined by finitely many rational linear inequalities and so induces a rational pointed cone. We provide formulas for the extremal rays of this cone by generalizing analogous formulas recently proven by the author and Belkale for the more specific scenario when the larger group is a direct product of the smaller group with itself (with the diagonal embedding). New rays not covered by these formulas also appear, but these are rare and easily identifiable by a procedure which we provide. We identify our first extremal rays thanks to a rigidity property stemming from a generalization of a conjecture of Fulton, which we first prove. Finally, we answer the Saturation Conjecture for three new connected, simple, complex Lie groups: type D_5, type D_6, and type E_6 (the first of the simply-laced exceptional types). This conjecture concerns the aforementioned cone in the diagonal embedding case and predicts that the need for weakening the original question to the asymptotic one is not necessary assuming the Lie group is simple of simply-laced type. Our verification takes as inspiration the proof of saturation for type D_4 given by Kapovich, Kumar, and Millson. However, we introduce a computational tool coming from the Localization Theorem in order to more efficiently calculate the necessary cup products using a supercomputer. Furthermore, the scale of the D_6 and E_6 calculations necessitates the use of the extremal ray formulas in order to directly produce rays instead of relying on software to find rays from inequalities.Doctor of Philosoph