principal specialization of projective Schur functions

Question

Is there a nice/known formula for the (some) principal specialization of a projective (P or Q) Schur polynomial? That is, I am talking about a projective analogue of Stanley's hook content formula $$s_{\lambda} (1,q,\dots,q^{n-1}) = \prod_{(i,j) \in \lambda} \frac{1-q^{n+j-i}}{1-q^{\lambda_i + \lambda_j'-i-j+1}}.$$

Answer 1

There is a very nice specialization when you consider infinitely many variables, due to an identity of Kawanaka: $$Q_{\lambda}(1,q,q^2,\dots)=\prod_{i=1}^m \frac{(-1;q)_{\lambda_i}}{(q;q)_{\lambda_i}} \prod_{1\le i < j \le m} \frac{q^{\lambda_j}-q^{\lambda_i}}{1-q^{\lambda_i+\lambda_j}}$$ where $\lambda$ is a partition of length $m$.

For the case of finitely many variables the story is a little more complicated. In the ordinary Schur polynomial case we have an expression which is a ratio of two determinants and is singular at $(1,1,\dots,1)$, however a Vandermonde factorization helps us get rid of the singularity and obtain an expression in $q$, which is also meaningful when $q=1$.

The same thing can be done with Schur-Q functions. The specialization can be written as a multiple hypergeometric sum, which is singular at $q=1$, but we can find a transformation formula into a hypergeometric sum of Schlosser type which extends to $q=1$. It would be a pain to write out the identities themselves, but if you follow Rosengren's paper on the subject, "Schur Q-polynomials, multiple hypergeometric series and enumeration of marked shifted tableaux", you should have what you want. Combinatorially this is interesting because it gives the $q$-enumeration of marked shifted tableaux.