Abstract
In this paper, we consider cyclotomic association schemes \(S = \mathrm {Cyc}(p^a, d)\). We focus on the adjacency algebra of S over algebraically closed fields K of characteristic p. If \(p\equiv 1 \pmod {d}\), \(p\equiv -1\pmod {d}\), or \(d\in \{2,3,4,5,6\}\), we identify the adjacency algebra of S over K as a quotient of a polynomial ring over an admissible ideal. In several cases, we determine the indecomposable direct sum decomposition of the standard module of S. As a consequence, we are able to compute the p-rank of several specific elements of the adjacency algebra of S over K.
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This work was supported by JSPS KAKENHI Grant Number 25400011.
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Hanaki, A. Modular adjacency algebras, standard representations, and p-ranks of cyclotomic association schemes. J Algebr Comb 44, 587–602 (2016). https://doi.org/10.1007/s10801-016-0681-y
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DOI: https://doi.org/10.1007/s10801-016-0681-y