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Computing with rational symmetric functions and applications to invariant theory and PI-algebras. (English) Zbl 1374.13009

Let \(K[[X]]\) be the algebra of formal power series over a field \(K\) in the variables \(X=\{x_1,\ldots,x_d\}\), and denote by \(K[[X]]^{S_d}\) its subalgebra of the symmetric functions. Then every \(f\in K[[X]]^{S_d}\) can be represented as \(f= \sum m(\lambda) S_\lambda(x_1,\ldots,x_d)\) where \(m(\lambda)\in K\) and \(S_\lambda\) are the Schur functions. The authors consider the so-called nice rational symmetric functions; these are \(f(X)\in K[[X]]^{S_d}\) that can be represented as rational functions with denominators that are products of elements of the type \(1-X^a = 1-x_1^{a_1}\cdots x_d^{a_d}\). Nice symmetric functions are rather important in invariant theory, and also in the theory of algebras with polynomial identities.
If \(f(X)\) is a symmetric function one denotes by \(M(f;X)=\sum m(\lambda)X^\lambda\) its multiplicity series. The first main result (Theorem 1.2) of the paper under review consists in exhibiting an expression for the multiplicity series of \(f(X)\). This is achieved by means of using and further developing classical methods and ideas of E. B. Elliott [Quart. J. 34, 348–377 (1902; JFM 34.0219.01)] and P. A. MacMahon [Combinatory analysis. Vol. 1 and 2 Cambridge: University Press (1915; JFM 45.1271.01; JFM 46.0118.07)] (the so-called \(\Omega\)-calculus). When \(f\) is a nice function the authors describe an algorithm for computing \(M(f;X)\). The algorithm looks pretty nice in the case of two variables but can also be useful for many variables. (Note that these results are, in essence, characteristic-free.)
The authors also study polynomial \(\mathrm{GL}_d\)-modules \(W\) in characteristic 0. It is well known that the Hilbert series of the symmetric algebra \(K[W]\) of \(W\) is a nice rational function. If one knows the multiplicity series of the latter Hilbert series then one can recover the decomposition of \(K[W]\) into a direct sum of irreducible \(\mathrm{GL}_d\)-modules. There are only a few instances where the Hilbert series of \(K[W]\) are known; the authors show how to compute the multiplicity series in these cases. Note that they include all \(W\) with \(\dim W<8\) and several modules \(W\) of dimension 8.
Moreover the paper surveys several interesting and important results concerning the invariants of the special linear group \(\mathrm{SL}_d\). The problem the authors are interested in is the following. Let \(W\) be a polynomial \(\mathrm{GL}_d\)-module, compute the Hilbert series of the algebra of invariants \(K[W]^G\) where \(G=\mathrm{SL}_d\), the special linear group (viewed as a subgroup of \(\mathrm{GL}_d\)), or \(G=\mathrm{UT}_d\), the unitriangular subgroup of \(\mathrm{GL}_d\). The authors produce such a method (Theorem 3.2) and further give several concrete examples.
The invariants of \(\mathrm{UT}_2\) can be described by means of the Weitzenböck derivations \(\delta\) (that is linear locally nilpotent derivations on the polynomial algebra \(K[Y]\)). The authors describe, in Theorem 3.5, the Hilbert series of the algebra of constants \(K[Y]^\delta\).
In the last section, the authors discuss applications of their methods to noncommutative invariant theory, and to the study of the numerical invariants of PI algebras. The paper ends with an extensive list of literature related to the topic (76 titles).

MSC:

13A50 Actions of groups on commutative rings; invariant theory
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
15A72 Vector and tensor algebra, theory of invariants
16R30 Trace rings and invariant theory (associative rings and algebras)
20G05 Representation theory for linear algebraic groups
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