An Orthosymplectic Pieri Rule

The classical Pieri formula gives a combinatorial rule for decomposing the product of a Schur function and a complete homogeneous symmetric polynomial as a linear combination of Schur functions with integer coefficients. We give a Pieri rule for describing the product of an orthosymplectic character and an orthosymplectic character arising from a one-row partition. We establish that the orthosymplectic Pieri rule coincides with Sundaram's Pieri rule for symplectic characters and that orthosymplectic characters and symplectic characters obey the same product rule.

On arithmetically realizable classes

We fix a number field L and a finite group G, and write Cl (ℤL[G]) for the reduced Grothendieck group of the category of finitely generated projective ℤL[G]-modules. We let RG denote the ring of complex characters of G, with SG the additive subgroup which is generated by the irreducible symplectic characters. We shall say that an element c ∈ Cl (ℤL[G]) is ‘(arithmetically) realizable’ if there exists a tamely ramified Galois extension N/K of number fields with L ⊆ K and an identification Gal (N/K) →˜ G via which c is the class of some Gal (N/K)-stble ℤN-ideal. We let RL(G) denote the subgroup of Cl (ℤL[G]) which is generated by the realizable elements for varying N/K. Our interest in RL(G) arises from the fact that it is the largest subset of Cl (ℤL[G]) upon which the results of Chinburg and the author in [Bu, Ch] can be used to give an explicit module theoretic description of the action of the integral semi-group ring AL, G of the Adams-Cassou-Noguès-Taylor operators (ΨL, k): k ∈ ℤ, 2 × k if SG ≠ {0}}. Whilst the results of [Bu, Ch] can (at least partially) be understood ‘geometrically’ via the action of Bott cannibalistic elements on suitable Grothendieck groups (cf. [Ch, E, P, T], [Bu]), the underlying problem of finding an explicit module theoretic interpretation of the action of AL, G on all elements of Cl(ℤL[G]) is of course essentially algebraic in nature. It is in this context that we were originally motivated to investigate RL(G).