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Search: All articles in the CJM digital archive with keyword Invariant theory

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1. CJM Online first

Klep, Igor; Špenko, Špela
Free function theory through matrix invariants
This paper concerns free function theory. Free maps are free analogs of analytic functions in several complex variables, and are defined in terms of freely noncommuting variables. A function of $g$ noncommuting variables is a function on $g$-tuples of square matrices of all sizes that respects direct sums and simultaneous conjugation. Examples of such maps include noncommutative polynomials, noncommutative rational functions and convergent noncommutative power series. In sharp contrast to the existing literature in free analysis, this article investigates free maps \emph{with involution} -- free analogs of real analytic functions. To get a grip on these, techniques and tools from invariant theory are developed and applied to free analysis. Here is a sample of the results obtained. A characterization of polynomial free maps via properties of their finite-dimensional slices is presented and then used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are series of generalized polynomials for which an invariant-theoretic characterization is given. Furthermore, an inverse and implicit function theorem for free maps with involution is obtained. Finally, with a selection of carefully chosen examples it is shown that free maps with involution do not exhibit strong rigidity properties enjoyed by their involution-free counterparts.

Keywords:free algebra, free analysis, invariant theory, polynomial identities, trace identities, concomitants, analytic maps, inverse function theorem, generalized polynomials
Categories:16R30, 32A05, 46L52, 15A24, 47A56, 15A24, 46G20

2. CJM 2004 (vol 56 pp. 495)

Gomi, Yasushi; Nakamura, Iku; Shinoda, Ken-ichi
Coinvariant Algebras of Finite Subgroups of $\SL(3,C)$
For most of the finite subgroups of $\SL(3,\mathbf{C})$, we give explicit formulae for the Molien series of the coinvariant algebras, generalizing McKay's formulae \cite{M99} for subgroups of $\SU(2)$. We also study the $G$-orbit Hilbert scheme $\Hilb^G(\mathbf{C}^3)$ for any finite subgroup $G$ of $\SO(3)$, which is known to be a minimal (crepant) resolution of the orbit space $\mathbf{C}^3/G$. In this case the fiber over the origin of the Hilbert-Chow morphism from $\Hilb^G(\mathbf{C}^3)$ to $\mathbf{C}^3/G$ consists of finitely many smooth rational curves, whose planar dual graph is identified with a certain subgraph of the representation graph of $G$. This is an $\SO(3)$ version of the McKay correspondence in the $\SU(2)$ case.

Keywords:Hilbert scheme, Invariant theory, Coinvariant algebra,, McKay quiver, McKay correspondence
Categories:14J30, 14J17

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