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Symmetric powers of \(\mathrm{Nat SL}(2,\mathbb{K})\). (English) Zbl 1368.20002

The main objective of the author is to study the symmetric powers of the natural representation of \(\text{SL}_2(\mathbb K)\) thought of as the spaces of homogeneous polynomials in two variables with fixed degree, among \(\mathbb Z[\text{SL}_2(\mathbb K)]\)-modules (\(\mathbb K\) a field).
Let \(\text{Sym}^k\text{Nat}\text{SL}_2(\mathbb Z)\) be the \(k\)-th symmetric power of the natural module \(\text{Nat}\text{SL}_2(\mathbb Z)=\mathbb Z^2\). In the first part of the article, the author investigates the tensor powers of the \(\mathbb Q[\text{SL}_2(\mathbb Z)]\)-modules \(\mathbb Q\otimes_{\mathbb Z}\text{Sym}^k\text{Nat}\text{SL}_2(\mathbb Z)=\text{Sym}_{\mathbb Q}^k\text{Nat}\text{SL}_2(\mathbb Z)\). This is the \((k+1)\)-dimensional space spanned by the vectors \(X^k,\;X^{k-1}Y,\;\dots,\;X Y^{k-1},\;Y^k\) over \(\mathbb Q\) and endowed with the usual action of \(\text{SL}_2(\mathbb Z)\leq\text{SL}_2(\mathbb Q)\) on polynomials. Given an \(\text{SL}_2(\mathbb Z)\)-module \(V\), the (nilpotent) length of \(V\) is the least integer \(k\) such that if \(u=\begin{pmatrix} 1&1\\0&1\end{pmatrix}\), then \((u-1)^k\cdot V=0\), i.e. \((u-1)^k=0\) in the endomorphism algebra of \(V\), if such \(k\) exists. The first main result states that if \(V\) has length at most \(5\), then V has a composition series each factor of which is a direct sum of copies of \(\text{Sym}_{\mathbb Q}^k\text{Nat}\text{SL}_2(\mathbb Z)\) for \(k=0,\dots,4\).
Then the author moves on to \(\text{SL}_2(\mathbb K)\) more generally and replaces \(u\) with \(u_\lambda=\begin{pmatrix} 1&\lambda\\0&1\end{pmatrix}\), for \(0\neq\lambda\in\mathbb K\).
Given a field \(\mathbb K\), let \(\mathbb K_1\subseteq\mathbb K\) denote its prime field and \(\overline{\mathbb K}\) its algebraic closure. We say that \(\mathbb K\) is \(k\)-radically closed if for any \(a\in\overline{\mathbb K}\) such that \(a^k\in\mathbb K\), then \(a\in\mathbb K\). For a positive integer \(n\), put \(n!!=n(n-2)(n-4)\cdots\big(n-2(\lceil\frac n2\rceil-1)\big)\). The second main theorem asserts that if \(\mathbb K\) has characteristic zero or at least \(2n+1\) and is \(2(n-1)!!\)-radically closed, and if \(V\) is an \(\text{SL}_2(\mathbb K)\)-module which is also a \(\mathbb K_1\)-vector space such that \(V\simeq\bigoplus_I\text{Sym}\text{Nat}\text{SL}_2(\mathbb K_1)\) as \(\mathbb K_1[\text{SL}_2(\mathbb K_1)]\)-modules, then \(V\) bears a compatible \(\mathbb K\)-vector space structure for which one has \(V\simeq\bigoplus_J\text{Sym}^{n-1}\text{Nat}\text{SL}_2(\mathbb K)\) as \(\mathbb K[\text{SL}_2(\mathbb K)]\)-modules. Here, \(I\) and \(J\) are some indexing sets.

MSC:

20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
20G15 Linear algebraic groups over arbitrary fields
20G05 Representation theory for linear algebraic groups
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[1] Andersen H. H., Jørgensen J. and Landrock P., The projective indecomposable modules of \({{\rm SL}(2,p^n)}\), Proc. Lond. Math. Soc. (3) 46 (1983), no. 1, 38-52.; Andersen, H. H.; Jørgensen, J.; Landrock, P., The projective indecomposable modules of \({{\rm SL}(2,p^n)}\), Proc. Lond. Math. Soc. (3), 46, 1, 38-52 (1983) · Zbl 0503.20013
[2] Cherlin G. and Deloro A., Small representations of \({\operatorname{SL}_2}\) in the finite Morley rank category, J. Symb. Log. 77 (2012), no. 3, 919-933.; Cherlin, G.; Deloro, A., Small representations of \({\operatorname{SL}_2}\) in the finite Morley rank category, J. Symb. Log., 77, 3, 919-933 (2012) · Zbl 1267.20053
[3] Deloro A., Changes to a theorem of Timmesfeld. I: Quadratic actions, Confluentes Math. 5 (2013), no. 2, 23-41.; Deloro, A., Changes to a theorem of Timmesfeld. I: Quadratic actions, Confluentes Math., 5, 2, 23-41 (2013) · Zbl 1327.20049
[4] Deloro A., Symmetric powers of \({\operatorname{Nat}\mathfrak{sl}_2(\mathbb{K})} \), Comm. Algebra (2016), 10.1080/ 00927872.2014.900690.; Deloro, A., Symmetric powers of \({\operatorname{Nat}\mathfrak{sl}_2(\mathbb{K})} \), Comm. Algebra (2016) · Zbl 1402.17017 · doi:10.1080/
[5] Grüninger M., On cubic action of a rank one group, preprint 2011, .; Grüninger, M., On cubic action of a rank one group (2011) · Zbl 1520.20001
[6] Lyndon R. C. and Schupp P. E., Combinatorial Group Theory, Classics Math., Springer, Berlin, 2001.; Lyndon, R. C.; Schupp, P. E., Combinatorial Group Theory (2001) · Zbl 0997.20037
[7] Serre J.-P., Trees, Springer Monogr. Math., Springer, Berlin, 2003.; Serre, J.-P., Trees (2003)
[8] Smith S. D., Quadratic action and the natural module for \({{\rm SL}_2(k)}\), J. Algebra 127 (1989), no. 1, 155-162.; Smith, S. D., Quadratic action and the natural module for \({{\rm SL}_2(k)}\), J. Algebra, 127, 1, 155-162 (1989) · Zbl 0688.20023
[9] Timmesfeld F. G., Abstract Root Subgroups and Simple Groups of Lie Type, Monogr. Math. 95, Birkhäuser, Basel, 2001.; Timmesfeld, F. G., Abstract Root Subgroups and Simple Groups of Lie Type (2001) · Zbl 0984.20019
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