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INVARIANT RINGS AND REPRESENTATIONS OF SYMMETRIC GROUPS

  • Kudo, Shotaro
  • Received : 2012.08.03
  • Published : 2013.07.31

Abstract

The center of the Lie group $SU(n)$ is isomorphic to $\mathbb{Z}_n$. If $d$ divides $n$, the quotient $SU(n)/\mathbb{Z}_d$ is also a Lie group. Such groups are locally isomorphic, and their Weyl groups $W(SU(n)/\mathbb{Z}_d)$ are the symmetric group ${\sum}_n$. However, the integral representations of the Weyl groups are not equivalent. Under the mod $p$ reductions, we consider the structure of invariant rings $H^*(BT^{n-1};\mathbb{F}_p)^W$ for $W=W(SU(n)/\mathbb{Z}_d)$. Particularly, we ask if each of them is a polynomial ring. Our results show some polynomial and non-polynomial cases.

Keywords

invariant theory;unstable algebra;pseudoreflection group;Lie group;p-compact group;classifying space

References

  1. K. K. S. Andersen and J. Grodal, The classification of 2-compact groups, J. Amer. Math. Soc. 22 (2009), no. 2, 387-436
  2. M. Craig, A characterization of certain extreme forms, Illinois J. Math. 20 (1976), no. 4, 706-717.
  3. W. G. Dwyer and C. W. Wilkerson, Kahler differentials, the T-functor, and a theorem of Steinberg, Trans. Amer. Math. Soc. 350 (1998), no. 12, 4919-4930. https://doi.org/10.1090/S0002-9947-98-02373-3
  4. W. G. Dwyer and C. W. Wilkerson, Poincare duality and Steinberg's theorem on rings of coinvariants, Proc. Amer. Math. Soc. 138 (2010), no. 10, 3769-3775. https://doi.org/10.1090/S0002-9939-2010-10429-X
  5. K. Ishiguro, Projective unitary groups and K-theory of classifying spaces, Fukuoka Univ. Sci. Rep. 28 (1998), no. 1, 1-6.
  6. K. Ishiguro, Invariant rings and dual representations of dihedral groups, J. Korean Math. Soc. 47 (2010), no. 2, 299-309. https://doi.org/10.4134/JKMS.2010.47.2.299
  7. R. M. Kane, Reflection Groups and Invariant Theory, CMS Books in Mathematics/Ouvrages de Mathematiques de la SMC, 5. Springer-Verlag, 2001.
  8. L. Smith, Polynomial Invariants of Finite Groups, A. K. Peters, Ltd., Wellesley, MA, 1995.