$C_p$-Mackey functors in Macaulay2
We introduce the CpMackeyFunctors package for Macaulay2, which allows for computations with Mackey functors over a cyclic group of prime order.
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arXiv subjects
Chenglu Wang
Publications and source records attributed to Chenglu Wang.
Representation stability in the (co)homology of vertical configuration spaces
In this paper, we study sequences of topological spaces called "vertical configuration spaces" of points in Euclidean space. We apply the theory of FI$_G$-modules, and results of Bianchi-Kranhold, to show that their (co)homology groups are "representation stable" with respect to natural actions of wreath products $S_k \wr S_n$. In particular, we show that in each (co)homological degree, the (co)homology groups (viewed as $S_k \wr S_n$-representations) can be expressed as induced representations of a specific form. Consequently, the characters of their rational (co)homology groups, and the patterns of irreducible $S_k \wr S_n$-representation constituents of these groups, stabilize in a strong sense. In addition, we give a new proof of rational (co)homological stability for unordered vertical configuration spaces, with an improved stable range.
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On Classifying Extensions of $p$-adic Fields
Let $p$ be a prime and let $\mathbb{Q}_p$ be the field of $p$-adic numbers. It is known that the finite extensions of $\mathbb{Q}_p$ of a given degree are finite up to isomorphism. Given a cubic field extension $L$ of $\mathbb{Q}_p$ generated by the root of an irreducible polynomial $h$, we present a practical (closed-form) method to determine the isomorphism class in which $L$ lives, based on the coefficients of $h$. We discuss the subtleties of the wildly ramified case, when the degree of the extension coincides with $p$, the characteristic of the residue field. We also present a method for tamely ramified extensions of arbitrary prime degree.
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