Extensions between Verma modules for dihedral groups

Computing the extensions between Verma modules is in general a very difficult problem. Using Soergel bimodules, one can construct a graded version of the principal block of Category $\mathcal{O}$ for any finite coxeter group. In this setting, we compute the extensions between Verma modules for dihedral groups

The regularity lemma is false over small fields

The regularity lemma is a stringent condition of the possible ranks of tensor blow-ups of linear subspaces of matrices. It was proved by Ivanyos, Qiao and Subrahmanyam when the underlying field is sufficiently large. We show that if the field size is too small, the regularity lemma is false

Polynomial degree bounds for matrix semi-invariants

We study the left-right action of $\operatorname{SL}_n \times \operatorname{SL}_n$ on $m$-tuples of $n \times n$ matrices with entries in an infinite field $K$. We show that invariants of degree $n^2- n$ define the null cone. Consequently, invariants of degree $\leq n^6$ generate the ring of invariants if $\operatorname{char}(K)=0$. We also prove that for $m \gg 0$, invariants of degree at least $n\lfloor \sqrt{n+1}\rfloor$ are required to define the null cone. We generalize our results to matrix invariants of $m$-tuples of $p\times q$ matrices, and to rings of semi-invariants for quivers. For the proofs, we use new techniques such as the regularity lemma by Ivanyos, Qiao and Subrahmanyam, and the concavity property of the tensor blow-ups of matrix spaces. We will discuss several applications to algebraic complexity theory, such as a deterministic polynomial time algorithm for non-commutative rational identity testing, and the existence of small division-free formulas for non-commutative polynomials.Comment: 16 page

Algorithms for orbit closure separation for invariants and semi-invariants of matrices

We consider two group actions on $m$-tuples of $n \times n$ matrices. The first is simultaneous conjugation by $\operatorname{GL}_n$ and the second is the left-right action of $\operatorname{SL}_n \times \operatorname{SL}_n$. We give efficient algorithms to decide if the orbit closures of two points intersect. We also improve the known bounds for the degree of separating invariants in these cases.Comment: Better bounds for separating invariants and improved exposition in some section

Degree bounds for semi-invariant rings of quivers

We use recent results on matrix semi-invariants to give degree bounds on generators for the ring of semi-invariants for quivers with no oriented cycles.Comment: 11 page

Explicit tensors of border rank at least 2d−22d-2 in Kd⊗Kd⊗KdK^d \otimes K^d \otimes K^d in arbitrary characteristic

For tensors in $\mathbb{C}^d \otimes \mathbb{C}^d \otimes \mathbb{C}^d$, Landsberg provides non-trivial equations for tensors of border rank $2d-3$ for $d$ even and $2d-5$ for $d$ odd were found by Landsberg. In previous work, we observe that Landsberg's method can be interpreted in the language of tensor blow-ups of matrix spaces, and using concavity of blow-ups we improve the case for odd $d$ from $2d-5$ to $2d-4$. The purpose of this paper is to show that the aforementioned results extend to tensors in $K^d \otimes K^d \otimes K^d$ for any field $K$.Comment: 12 page

On non-commutative rank and tensor rank

We study the relationship between the commutative and the non-commutative rank of a linear matrix. We give examples that show that the ratio of the two ranks comes arbitrarily close to 2. Such examples can be used for giving lower bounds for the border rank of a given tensor. Landsberg used such techniques to give nontrivial equations for the tensors of border rank at most $2m-3$ in $K^m\otimes K^m\otimes K^m$ if $m$ is even. He also gave such equations for tensors of border rank at most $2m-5$ in $K^m\otimes K^m\otimes K^m$ if $m$ is odd. Using concavity of tensor blow-ups we show non-trivial equations for tensors of border rank $2m-4$ in $K^m \otimes K^m \otimes K^m$ for odd $m$ for any field $K$ of characteristic 0. We also give another proof of the regularity lemma by Ivanyos, Qiao and Subrahmanyam.Comment: 15 page

An exponential lower bound for the degrees of invariants of cubic forms and tensor actions

Using the Grosshans Principle, we develop a method for proving lower bounds for the maximal degree of a system of generators of an invariant ring. This method also gives lower bounds for the maximal degree of a set of invariants that define Hilbert's null cone. We consider two actions: The first is the action of ${\rm SL}(V)$ on ${\rm Sym}^3(V)^{\oplus 4}$, the space of $4$-tuples of cubic forms, and the second is the action of ${\rm SL}(V) \times {\rm SL}(W) \times {\rm SL}(Z)$ on the tensor space $(V \otimes W \otimes Z)^{\oplus 9}$. In both these cases, we prove an exponential lower degree bound for a system of invariants that generate the invariant ring or that define the null cone

Highly entangled tensors

A geometric measure for the entanglement of a unit length tensor $T \in (\mathbb{C}^n)^{\otimes k}$ is given by $- 2 \log_2 ||T||_\sigma$, where $||.||_\sigma$ denotes the spectral norm. A simple induction gives an upper bound of $(k-1) \log_2(n)$ for the entanglement. We show the existence of tensors with entanglement larger than $k \log_2(n) - \log_2(k) - o(\log_2(k))$. Friedland and Kemp have similar results in the case of symmetric tensors. Our techniques give improvements in this case.Comment: 13 pages, improved results and a section added on symmetric tensor

Maximum likelihood estimation for matrix normal models via quiver representations

In this paper, we study the log-likelihood function and Maximum Likelihood Estimate (MLE) for the matrix normal model for both real and complex models. We describe the exact number of samples needed to achieve (almost surely) three conditions, namely a bounded log-likelihood function, existence of MLEs, and uniqueness of MLEs. As a consequence, we observe that almost sure boundedness of log-likelihood function guarantees almost sure existence of an MLE, thereby proving a conjecture of Drton, Kuriki and Hoff. The main tools we use are from the theory of quiver representations, in particular, results of Kac, King and Schofield on canonical decomposition and stability.Comment: 26 page