# Does the ring generated by the odd power sum symmetric functions have a name?

Let $\Lambda$ be the ring of symmetric functions and recall the power sum symmetric function $p_i = \sum x_1^i + x_2^i + \dots$ generate this ring. Let $\tilde\Lambda$ be the ring generated by the odd power sums. Then $\tilde\Lambda$ has several interesting properties: it satisfies the $Q$-cancellation property $$f(x_1,-x_1,x_2,x_3,\dots) = f(x_2,x_3,\dots),$$ has the $Q$-Schur and $P$-Schur functions as orthogonal bases and is closely related to the theory of projective representations for the symmetric group.

As best I can tell (after having asked several experts), $\tilde\Lambda$ lacks a simple name. I have seen/heard people refer to it as the ring of symmetric functions generated by the odd power sums or the ring of symmetric functions satisfying the $Q$-cancellation property, but given its prominence I would have expected a name like the __________ symmetric functions. Is anyone aware of such a name? If someone can propose an informative name, I would be happy to start referring to $\tilde\Lambda$ as such in my future work.

• Maybe create a name yourself? Aug 1, 2015 at 14:01
• @Ryan I've been tempted. Aug 1, 2015 at 18:01

Over a field of characteristic $0$, this is the odd subalgebra of $\left(Sym, \zeta_S\right)$, in the notation of Marcelo Aguiar, Nantel Bergeron, and Frank Sottile, Combinatorial Hopf algebras and generalized Dehn-Sommerville equations (this link goes to a corrected postprint).