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Register a new userSymmetric powers of Nat SL(2,K)
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Authors
Adrien Deloro
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We identify the representations $mathbb{K}[X^k, X^{k-1}Y, dots, Y^k]$ among abstract $mathbb{Z}[mathrm{SL}_2(mathbb{K})]$-modules. One result is on $mathbb{Q}[mathrm{SL}_2(mathbb{Z})]$-modules of short nilpotence length and generalises a classical quadratic theorem by Smith and Timmesfeld. Another one is on extending the linear structure on the module from the prime field to $mathbb{K}$. All proofs are by computation in the group ring using the Steinberg relations.
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We identify the spaces of homogeneous polynomials in two variables K[Y^k, XY^{k-1}, ..., X^k] among representations of the Lie ring sl(2,K). This amounts to constructing a compatible K-linear structure on some abstract sl(2,K)-modules, where sl(2,K) is viewed as a Lie ring.
We show that an element w of a free group F on n generators defines a surjective word map of PSL(2,C)^n onto PSL(2,C) unless w belongs to the second derived subgroup of F. We also describe certain words maps that are surjective on SL(2,C) x SL(2,C). Here C is the field of complex numbers.
A superspace formulation of IIB supergravity which includes the field strengths of the duals of the usual physical one, three and five-form field strengths as well as the eleven-form field strength is given. The superembedding formalism is used to construct kappa-symmetric SL(2,R) covariant D-brane actions in an arbitrary supergravity background.
In algebraic terms, the insertion of $n$-powers in words may be modelled at the language level by considering the pseudovariety of ordered monoids defined by the inequality $1le x^n$. We compare this pseudovariety with several other natural pseudovarieties of ordered monoids and of monoids associated with the Burnside pseudovariety of groups defined by the identity $x^n=1$. In particular, we are interested in determining the pseudovariety of monoids that it generates, which can be viewed as the problem of determining the Boolean closure of the class of regular languages closed under $n$-power insertions. We exhibit a simple upper bound and show that it satisfies all pseudoidentities which are provable from $1le x^n$ in which both sides are regular elements with respect to the upper bound.
In this paper we study the (2,k)-generation of the finite classical groups SL(4,q), Sp(4,q), SU(4,q^2) and their projective images. Here k is the order of an arbitrary element of SL(2,q), subject to the necessary condition k>= 3. When q is even we allow also k=4.
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