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Some modular representations of reflection groups related to Weyl groups are considered. The rational cohomology of the classifying space of a compact connected Lie group $G$ with a maximal torus $T$ is expressed as the ring of invariants, $H^*(BG; \Q)\cong H^*(BT; \Q)^{W(G)}$, which is a polynomial ring. If such Lie groups are locally isomorphic, the rational representations of their Weyl groups are equivalent. However, the integral representations need not be equivalent. Under the mod $p$ reductions, we consider the structure of the rings, particularly for the Weyl group of symplectic groups $Sp(n)$ and for the alternating groups $A_n$ as the subgroup of $W(SU(n))$. We will ask if such rings of invariants are polynomial rings, and if each of them can be realized as the mod $p$ cohomology of a space. For $n=3, 4$, the rings under a conjugate of $W(Sp(n))$ are shown to be polynomial, and for $n=6, 8$, they are non--polynomial. The structures of $H^*(BT^{n-1}; \F_p)^{A_n}$ will be also discussed for $n=3, 4$.
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참고문헌 (21)
참고문헌 신청
K. K. S. Andersen / 2008 / The Steenrod problem of realizing polynomial cohomology rings / J. Topol 1(4):747~760
Kasper K. S. Andersen / 2009 / The classification of $2$–compact groups / Journal of the American Mathematical Society 22(2):387~436
A. Clark / 1974 / The realization of polynomial algebras as cohomology rings / Pa-cific J. Math 50:425~434
H. Duan / 2017 / The cohomology of projective unitary groups, talk / The 2nd PanPacific International Conference on Topology and Applications at Busan
W.G. Dwyer / 1992 / Homotopical uniqueness of classifying spaces / Topology 31(1):29~45
Kasper K. S. Andersen / 2009 / The classification of $2$–compact groups / Journal of the American Mathematical Society 22(2):387~436
A. Clark / 1974 / The realization of polynomial algebras as cohomology rings / Pa-cific J. Math 50:425~434
H. Duan / 2017 / The cohomology of projective unitary groups, talk / The 2nd PanPacific International Conference on Topology and Applications at Busan
W.G. Dwyer / 1992 / Homotopical uniqueness of classifying spaces / Topology 31(1):29~45
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