arXiv subjects

Robert A. Spencer

Publications and source records attributed to Robert A. Spencer.

3 recordsLinked to original sources

arXiv2026

Cell Modules for Type $A$ Webs

We examine the cell modules for the category of type An webs and their natural cellular forms. We modify the bases of these modules, as described by Elias, to obtain an orthogonal basis of each cell module. Hence, we calculate the determinant of the Gram matrix with respect to such bases. These Gram determinants are given in terms of intersection forms, computed from certain traces of clasps - higher order Jones-Wenzl morphisms. Additionally, the modified basis is constructed using these clasps, and each clasp is constructed using traces of smaller clasps. Elias conjectures a value for these intersection forms and verifies it in types $A_1$, $A_2$ and $A_3$. This paper concludes with a proof of the conjecture in type $A_n$.

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arXiv2026

Cell modules for the Temperley-Lieb algebra in mixed characteristic

We study the representation theory of the Temperley-Lieb algebra $\mathsf{TL}_n^k(δ)$ in mixed characteristic, i.e. over an arbitrary field $k$ of characteristic $p$ and where $δ$ satisfies some minimal polynomial $m_δ$. In particular, we completely describe the submodule structure of cell modules for $\mathsf{TL}_n$ and give their Alperin diagrams. The proof is entirely diagrammatic and does not appeal to the role of $\mathsf{TL}_n$ as the endomorphism algebra of tensor powers of the fundamental representation of $\textbf{U}_q(\mathfrak{sl}_2)$. We also investigate two-dimensional Jantzen-like filtrations of the cell modules related to the mixed characteristic.

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arXiv2021

Non-induced modular representations of cyclic groups

We compute the ring of non-induced representations for a cyclic group, $C_n$, over an arbitrary field and show that it has rank $φ(n)$, where $φ$ is Euler's totient function - independent of the characteristic of the field. Along the way, we obtain a "pick-a-number" trick; expressing an integer $n$ as a sum of products of $p$-adic digits of related integers.

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