Coinduction functor in representation stability theory

Internal zonotopal algebras and the monomial reflection groups G(m,1,n)

“…Based on our theorem above, it is perhaps unsurprising that the Lie representation appears in type A, however our perspective makes obvious the "hidden" action of a larger symmetric group than is necessary on the Lie representation, as observed by Mathieu [18] and Robinson and Whitehouse [22]. For all G(m, 1, n), we give explicit generators of the internal zonotopal ideal and then use this description to prove finite generation in the sense of Sam and Snowden [25] and representation stability as described by Gan and Li [10]. We generalize a recurrence relation for the Whitehouse representation which factorizes the regular representation of G(m, 1, n), m > 1.…”

On the negative-one shift functor for FI-modules

“…Immediate from Theorem 9 and the definition of Q ′ . From Corollary 10 and (1), we recover [3,Theorem 1.3] for FI.…”

“…The coinduction functor Q on the category of FI-modules was defined in [3, Definition 4.1] as S † . It is a right adjoint functor of S by Proposition 2 (see [3,Lemma 4.2]). In this section, we give an explicit description of Q in terms of S −1 .…”

“…Remark 11. In [3], for any finite field F q , Gan and Li also studied the coinduction functor for VI-modules where VI is the category whose objects are the finite dimensional vector spaces over F q and whose morphisms are the injective linear maps. Similarly to FI-modules, the coinduction functor for VI-modules is defined as S † where S is the shift functor for VImodules.…”

“…The modest goal of this article is to explain how they are related to the negative-one shift functor S −1 introduced in [2]. We also explain how the coinduction functor Q defined in [3] is related to S −1 .…”