Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Free functions with symmetry   In 1936, Margarete C. Wolf showed that the ring of symmetric free polynomials
in two or more variables is isomorphic to the ring of free polynomials in
infinitely many variables. We show that Wolf's theorem is a special case of a
general theory of the ring of invariant free polynomials: every ring of
invariant free polynomials is isomorphic to a free polynomial ring.
Furthermore, we show that this isomorphism extends to the free functional
calculus as a norm-preserving isomorphism of function spaces on a domain known
as the row ball. We give explicit constructions of the ring of invariant free
polynomials in terms of representation theory and develop a rudimentary theory
of their structures. Specifically, we obtain a generating function for the
number of basis elements of a given degree and explicit formulas for good bases
in the abelian case.

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The categories of this paper are: q-fin
35
The categories of this paper are: mtrl-th
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The categories of this paper are: econ
22
The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: funct-an
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
21
The categories of this paper are: adap-org