Geodesic
Operations in Grothendieck Rings and the Symmetric Group
Canadian journal of mathematics, Tome 26 (1974) no. 3, pp. 543-550

Voir la notice de l'article provenant de la source Cambridge University Press

In [1] Atiyah described how to use the complex representations of the symmetric group, Sn , to define and investigate operations in complex topological K-theory. In this paper operations for more general Grothendieck groups are described in terms of the integral representations of Sn using the representations directly without passing to the dual as Atiyah did. The principal tool, which is proved in the first section, is the theorem that the direct sum of the Grothendieck groups of finite integral representations of Sn form a bialgebra isomorphic to a polynomial ring with a sequence of divided powers. A consequence of this theorem is that the only operations that can be constructed from the symmetric groups will be polynomials in the symmetric powers. 
Burroughs, John. Operations in Grothendieck Rings and the Symmetric Group. Canadian journal of mathematics, Tome 26 (1974) no. 3, pp. 543-550. doi: 10.4153/CJM-1974-050-3
@article{10_4153_CJM_1974_050_3,
     author = {Burroughs, John},
     title = {Operations in {Grothendieck} {Rings} and the {Symmetric} {Group}},
     journal = {Canadian journal of mathematics},
     pages = {543--550},
     year = {1974},
     volume = {26},
     number = {3},
     doi = {10.4153/CJM-1974-050-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1974-050-3/}
}
TY  - JOUR
AU  - Burroughs, John
TI  - Operations in Grothendieck Rings and the Symmetric Group
JO  - Canadian journal of mathematics
PY  - 1974
SP  - 543
EP  - 550
VL  - 26
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1974-050-3/
DO  - 10.4153/CJM-1974-050-3
ID  - 10_4153_CJM_1974_050_3
ER  - 
%0 Journal Article
%A Burroughs, John
%T Operations in Grothendieck Rings and the Symmetric Group
%J Canadian journal of mathematics
%D 1974
%P 543-550
%V 26
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1974-050-3/
%R 10.4153/CJM-1974-050-3
%F 10_4153_CJM_1974_050_3

[1] 1. Atiyah, M. F., Power operations in k-theory, Quart. J. Math. Oxford Ser. 17 (1966), 165–93. Google Scholar

[2] 2. Atiyah, M. F. and Tall, D. O., Group representations, λ-rings and the J-homomorphism, Topology 8 (1969), 253–97. Google Scholar

[3] 3. Berthelot, P., Grothendieck, A., and Illusie, L., Théorie des Intersections et Théorème de Riemann-Roch (SGA6), Lecture Notes in Math. 225 (Springer, Berlin, 1971). Google Scholar

[4] 4. Boerner, H., Representations of groups (North-Holland Publishing Co., Amsterdam, 1963). Google Scholar

[5] 5. Cartier, P., Groupes formels associés aux anneaux de Witt généralisés, C. R. Acad. Sci. Paris, Sér. A-B 265 (1967), 49–42. Google Scholar

[6] 6. Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras, (Interscience Publ., New York, 1962). Google Scholar

[7] 7. Heller, A. and Reiner, I., Grothendieck groups of orders in semi-simple algebras, Trans. Amer. Math. Soc. 112 (1964), 344–55. Google Scholar

[8] 8. Kerber, A., Representations of permutation groups. I, Lecture Notes in Mathematics 240 (Springer, Berlin, 1971). Google Scholar

[9] 9. Robinson, G. de B., Representation theory of the symmetric group (University of Toronto Press, Toronto 1961). Google Scholar

[10] 10. Swan, R., A splitting principle in algebraic K-theory, Representation theory of finite groups and related topics, Proc. Sym. in Pure Math., XXI (1971), 155–160. Google Scholar

Cité par Sources :