Location of Repository

Vector invariants for the two dimensional modular representation of a cyclic group of prime order

By Eddy Campbell, R. James Shank and David L. Wehlau


In this paper, we study the vector invariants, F[mV_2]^(C_p), of the 2-dimensional indecomposable representation V_2 of the cylic group, C_p, of order p over a field F of characteristic p. This ring of invariants was first studied by David Richman who showed that this ring required a generator of degree m(p-1), thus demonstrating that the result of Noether in characteristic 0 (that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group) does not extend to the modular case. He also conjectured that a certain set of invariants was a generating set with a proof in the case p=2. This conjecture was proved by Campbell and Hughes. Later, Shank and Wehlau determined which elements in Richman's generating set were redundant thereby producing a minimal generating set.\ud \ud We give a new proof of the result of Campbell and Hughes, Shank and Wehlau giving a minimal algebra generating set for the ring of invariants F[m V_2]^(C_p). In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis. Further, our techniques also serve to give an explicit decomposition of F[m V_2] into a direct sum of indecomposable C_p-modules. Finally, noting that our representation of C_p on V_2 is as the p-Sylow subgroup of SL_2(F_p), we are able to determine a generating set for the ring of invariants of F[m V_2]^(SL_2(F_p))

Topics: QA150
Year: 2010
OAI identifier: oai:kar.kent.ac.uk:23885

Suggested articles



  1. (1989). A completion procedure for computing a canonical basis of a k-subalgebra, doi
  2. (1911). A Fundamental System of Invariants of the General Modular Linear Group with a Solution of the Form Problem, doi
  3. (1983). A Primer on the Dickson Invariants, doi
  4. (1996). Bases for rings of coinvariants, doi
  5. (1998). bases for rings of formal modular seminvariants, doi
  6. (2008). Catalan Numbers with Application doi
  7. (2002). Computational invariant theory, Invariant Theory and Algebraic Transformation Groups, I, doi
  8. (2002). Computing modular invariants of p-groups, doi
  9. (1915). Der Endlichkeitssatz der invarianten endlicher Gruppen, doi
  10. (1954). Finite unitary reflection groups, doi
  11. (1967). Groupes finis d’automorphismes d’anneaux locaux r´ eguliers, Colloque d’Alg` ebre (Paris,
  12. (1967). Groupes finis d’automorphismes d’anneaux locaux re´guliers, Colloque d’Alge`bre (Paris,
  13. (1992). Ideals, varieties, and algorithms, doi
  14. (2002). Invariant theory of finite groups, doi
  15. (1955). Invariants of finite groups generated by reflections, doi
  16. Modular Invariant Theory, doi
  17. (2002). Noether numbers for subrepresentations of cyclic groups of prime order, doi
  18. (2001). On Noether’s bound for polynomial invariants of a finite group, doi
  19. On regular difference terms, doi
  20. (1990). On vector invariants over finite fields, doi
  21. (1995). Polynomial invariants of finite groups, doi
  22. (1980). Profondeur d’anneaux d’invariants en caract´ eristique p,
  23. (1980). Profondeur d’anneaux d’invariants en caracte´ristique p,
  24. (1990). Subalgebra bases, doi
  25. (2000). Symmetric powers of modular representations, Hilbert series and degree bounds, doi
  26. (1997). The classical groups, doi
  27. (1997). The Magma algebra system I: the user language, doi
  28. (2000). The Noether bound in invariant theory of finite groups, doi
  29. (1951). The product of the generators of a finite group generated by reflections,
  30. (1997). Vector invariants of U2(Fp): A proof of a conjecture of Richman, doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.