Browsing by Subject "Invariants."

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    Hyperdeterminants, entangled states, and invariant theory
    (2013) Şen, Emre
    In [1] and [2], A. Klyachko connects quantum entanglement and invariant theory so that entangled state of a quantum system can be explained by invariants of the system. After representing states in multidimensional matrices, this relation turns into finding multidimensional matrix invariants so called hyperdeterminants. Here we provide a necessary and sufficient condition for existence of a hyperdeterminant of a multidimensional matrix of an arbitrary format. The answer is given in terms of the so called castling transform that relates hyperdeterminants of different formats. Among castling equivalent formats there is a unique castling reduced one, that has minimal number of entries. We prove the following theorem: “Multidimensional matrices of a given format admit a non-constant hyperdeterminant if and only if logarithm of dimensions of the castling reduced format satisfy polygonal inequalities.”
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    Invariant rings of modular P-groups
    (2013) Toper, Ceren Coşkun
    We consider a finite group acting as linear substitutions on a polynomial ring and study the corresponding ring of invariants. Computing the invariant ring and finding its ring theoretical properties is a classical problem. We focus on the modular case where the characteristic of the field divides the order of the group. We review invariants of basic modular actions and give explicit descriptions of invariants of small dimensional actions. We also discuss a recent algorithm that computes the invariant ring of a modular p-group up to a localization and we apply this algorithm to invariants of indecomposable representations of a cyclic group of prime order.