λ-Rings and the Representation Theory of the Symmetric Group

Knutson, Donald M.

doi:10.1007/bfb0069217

Cited by 161 publications

(191 citation statements)

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“…Every virtual representation ring R(G) has the extra structure of a λ-ring. See [12] for the definitions and basic properties of λ-rings and pre-λ-rings. Here we just briefly state the most relevant facts.…”

Section: Proposition 1 (Homology Isomorphism)mentioning

confidence: 99%

An Additive Basis for the Chow Ring of M_0,2(P^r,2)

Cox¹

2007

SIGMA

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Abstract. We begin a study of the intersection theory of the moduli spaces of degree two stable maps from two-pointed rational curves to arbitrary-dimensional projective space. First we compute the Betti numbers of these spaces using Serre polynomial and equivariant Serre polynomial methods developed by E. Getzler and R. Pandharipande. Then, via the excision sequence, we compute an additive basis for their Chow rings in terms of Chow rings of nonlinear Grassmannians, which have been described by Pandharipande. The ring structure of one of these Chow rings is addressed in a sequel to this paper.

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Section: Proposition 1 (Homology Isomorphism)mentioning

confidence: 99%

An Additive Basis for the Chow Ring of M_0,2(P^r,2)

Cox¹

2007

SIGMA

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“…Finally, we remark that the tpm defined above are the "Adams operations" for a X-ring structure [5] on the Burnside rings of abelian groups. This should be contrasted with the (unpublished) result of E. Boorman that © (S3) admits no X-ring structure (contrary to a claim in [5]).…”

Section: We Call An Endomorphism Of ©(G) ® Q Integral If It Maps ©(G)mentioning

confidence: 99%

Natural endomorphisms of Burnside rings

Blass¹

1979

Trans. Amer. Math. Soc.

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Abstract. The Burnside ring $ (G) of a finite group G consists of formal differences of finite G-sets. 'S is a contravariant functor from finite groups to commutative rings. We study the natural endomorphisms of this functor, of its extension Q ® 'S to rational scalars, and of its restriction 'S f Ab to abelian groups. Such endomorphisms are canonically associated to certain operators that assign to each group one of its conjugacy classes of subgroups. Using these operators along with a carefully constructed system of linear congruences defining the image of ® (G) under its canonical embedding in a power of Z, we exhibit a multitude of natural endomorphisms of ®, we show that only two of them map G-sets to G-sets, and we completely describe all natural endomorphisms of <$> [ Ab.Finite sets on which a finite group G acts (G-sets) can be added and multiplied in an obvious way, so their formal differences (virtual G-sets) form a ring, the Burnside ring of G. This paper is concerned with natural endomorphisms of Burnside rings, i.e., operators 9 mapping virtual G-sets to virtual G-sets for all G and respecting addition, multiplication, and "restriction of scalars" along homomorphisms of groups. (Precise definitions are given in §1.) If we require 0 to send G-sets to G-sets (rather than to virtual G-sets), then, as we prove in §6, the only 9 other than the identity is the one that sends every G-set to the same set with G acting trivially. But, in the absence of this positivity requirement, the monoid of natural endomorphisms is uncountable (Corollary 4c), noncommutative (Corollary 4d), and apparently quite chaotic (see the end of §4). In contrast, we show in §5 that the natural endomorphisms of Burnside rings of abelian groups admit a uniform description and commute with each other. In the course of obtaining these results, we describe (in §3) natural endomorphisms 9 of rational Burnside algebras in terms of certain operators 9 that assign to each group a subgroup, and we give (in §4) a sufficient condition on 0 for 9 to be an endomorphism of Burnside rings. Although this sufficient condition is not known to be necessary, it covers all known examples.§1 contains definitions and well-known preliminary results, including a

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“…A G-module V is said to be decomposable if V = Vx II V2 as a G-module, where V¡ ¥= 0. Propositions 1.7-1.9 can be found in any book on group representation theory (see [5], [8] for…”

Section: -Operations 131mentioning

confidence: 99%

“…Unigeneration of the X-ring R(Sk). It is well known that R(Sk) is a free Z-module with basis {lndsk xsk2x-sk II */ > 1> 2/*/ = k} [5,Chapter III]. Therefore, P(ASk) = AT0 (A^k) = R(sJ, where A = finite-dimensional C. Some open questions.…”

Section: E H Boormanmentioning

confidence: 99%