11,211 research outputs found
Semisimplicity in symmetric rigid tensor categories
Let \lambda be a partition of a positive integer n. Let C be a symmetric
rigid tensor category over a field k of characteristic 0 or char(k)>n, and let
V be an object of C. In our main result (Theorem 4.3) we introduce a finite set
of integers F(\lambda) and prove that if the Schur functor \mathbb{S}_{\lambda}
V of V is semisimple and the dimension of V is not in F(\lambda), then V is
semisimple. Moreover, we prove that for each d in F(\lambda) there exist a
symmetric rigid tensor category C over k and a non-semisimple object V in C of
dimension d such that \mathbb{S}_{\lambda} V is semisimple (which shows that
our result is the best possible). In particular, Theorem 4.3 extends two
theorems of Serre for C=Rep(G), G is a group, and \mathbb{S}_{\lambda} V is
\wedge^n V or Sym^n V, and proves a conjecture of Serre (\cite{s1}).Comment: 15 pages, minor corrections in Subsection 4.6 and in the proof of
Lemma 4.2
Query-based Deep Improvisation
In this paper we explore techniques for generating new music using a
Variational Autoencoder (VAE) neural network that was trained on a corpus of
specific style. Instead of randomly sampling the latent states of the network
to produce free improvisation, we generate new music by querying the network
with musical input in a style different from the training corpus. This allows
us to produce new musical output with longer-term structure that blends aspects
of the query to the style of the network. In order to control the level of this
blending we add a noisy channel between the VAE encoder and decoder using
bit-allocation algorithm from communication rate-distortion theory. Our
experiments provide new insight into relations between the representational and
structural information of latent states and the query signal, suggesting their
possible use for composition purposes
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