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.

  • $\begingroup$ Maybe create a name yourself? $\endgroup$ – Ryan Aug 1 '15 at 14:01
  • $\begingroup$ @Ryan I've been tempted. $\endgroup$ – Zachary Hamaker Aug 1 '15 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).

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    $\begingroup$ Unless I am mistaken, the odd symmetric functions are by Khovanov and Ellis, and are not the ones that the OP mentioned. $\endgroup$ – user61318 Aug 1 '15 at 19:35
  • $\begingroup$ @user61318: Thank you! Yes, they're a totally different thing (and I should have known that, having read much of both papers). $\endgroup$ – darij grinberg Aug 1 '15 at 19:53

Darij gave a very good reference indeed. One other instance where I saw this before is the Maple package SF, where they talk about "signed class functions" (http://www.math.lsa.umich.edu/~jrs/software/QFhelp.html#char2qf), this might be also useful in choosing the name, if you are so inclined.


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