Adem relations in the Dyer-Lashof algebra and modular invariants.

*(English)*Zbl 1058.55007To 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)].

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)].

Reviewer: Michael Joachim (Münster)

##### 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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\textit{N. E. Kechagias}, Algebr. Geom. Topol. 4, 219--241 (2004; Zbl 1058.55007)

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