Given a partition λ of n, consider the subspace E λ of C n where the first λ 1 coordinates are equal, the next λ 2 coordinates are equal, etc. In this paper, we study subspace arrangements X λ consisting of the union of translates of E λ by the symmetric group. In particular, we focus on determining when X λ is Cohen-Macaulay. This is inspired by previous work of the third author coming from the study of rational Cherednik algebras and which answers the question positively when all parts of λ are equal. We show that X λ is not Cohen-Macaulay when λ has at least 4 distinct parts, and handle a large number of cases when λ has 2 or 3 distinct parts. Along the way, we also settle a conjecture of Sergeev and Veselov about the Cohen-Macaulayness of algebras generated by deformed Newton sums. Our techniques combine classical techniques from commutative algebra and invariant theory; in many cases we can reduce an infinite family to a finite check which can sometimes be handled by computer algebra.
For a coalgebra C k over field k, we define the "coalgebra extension problem" as the question: what multiplication laws can we define on C k to make it a bialgebra over k? This paper answers this existence-uniqueness question for certain special coalgebras. We begin with the trigonometric coalgebra, comparing and contrasting it with the group-(bi)algebra k[Z/2]. This leads to a generalization, the coalgebra dual to the group-algebra k[Z/p], which we then investigate. We show the connections with other problems, and see that the answer to the coalgebra extension problem for these families depends interestingly on the base field k.
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