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Adem relations in the Dyer-Lashof algebra and modular invariants. (English) Zbl 1058.55007
To set the stage, recall that the Dyer-Lashof algebra for a prime $$p>2$$ can be constructed in two steps: first one starts with the free graded associative algebra over the field $$F_p$$ with $$p$$ elements on generators $$\{f^i, i\geq 0\}$$ and $$\{ \beta f^i$$, $$i>0\}$$ of degrees $$2(p-1)i$$ and $$2i(p-1)-1$$, respectively, and divides out the ideal generated by elements of negative excess, giving a graded algebra $$T$$; the Dyer-Lashof algebra $$R$$ then is obtained from $$T$$ as the quotient of $$T$$ by the ideal generated by the Adem relations. The images of the generators $$f_i$$ and $$\beta f_i$$ in $$R$$ now correspond to the operations $$Q_i$$ and $$\beta Q_i$$. Let $$T^*$$ and $$R^*$$ denote the vector space duals of $$T$$ and $$R$$ respectively.
In the author’s paper [Manuscr. Math. 84, No. 3–4, 261–290 (1994; Zbl 0847.55012)] he defines explicit isomorphisms $$T^* \cong A$$ and $$R^*\cong B$$ which identify $$T^*$$ and $$R^*$$ with subalgebras $$A$$ and $$B$$ in $$H^*(B(\mathbb Z/p)^n;F_p)$$. The algebra $$B$$ is invariant with respect to the action of Gl$$_n(F_p)$$, while the algebra $$B$$ is invariant with respect to the action of the Borel subgroup $$B_n \subset \text{Gl}_n(F_p)$$. Moreover there is a canonical inclusion $$i: B \subset A$$. In the paper under review the author verifies that the inclusion $$R^* \subset T^*$$ and the inclusion $$B \subset A$$ are compatible with the isomorphisms in [loc. cit.], and he gives an algorithm which computes the isomorphism $$B \to R^*$$, using distinguished bases for $$B$$ and $$R^*$$, respectively. The author then uses these results to establish a second algorithm which computes the map $$R^* \to T^*$$ in terms of invariant theory. The latter naturally leads to an interpretation of the complexity of the Adem relations in $$R$$ from a modular invariant point of view. Corresponding results for the prime $$2$$ were obtained earlier by Huynn Mùi in [Math. Z. 193, 151–163 (1986; Zbl 0597.55019)].

##### MSC:
 55S12 Dyer-Lashof operations 13A50 Actions of groups on commutative rings; invariant theory 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 55S10 Steenrod algebra
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