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Modular invariants under the actions of some reflection groups related to Weyl groups
Kenshi Ishiguro,Takahiro Koba,Toshiyuki Miyauchi,Erika Takigawa 대한수학회 2020 대한수학회보 Vol.57 No.1
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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MODULAR INVARIANTS UNDER THE ACTIONS OF SOME REFLECTION GROUPS RELATED TO WEYL GROUPS
Ishiguro, Kenshi,Koba, Takahiro,Miyauchi, Toshiyuki,Takigawa, Erika Korean Mathematical Society 2020 대한수학회보 Vol.57 No.1
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) ≅ H*(BT; ℚ)<sup>W(G)</sup>, 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<sub>n</sub> 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<sup>n-1</sup>; 𝔽<sub>p</sub>)<sup>A<sub>n</sub></sup> will be also discussed for n = 3, 4.
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