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Tableau atoms and a new Macdonald positivity conjecture. (English) Zbl 1020.05069
A new family of symmetric polynomials $$A_{\lambda}^{(k)}[X;t]$$ labelled by partitions $$\lambda$$ such that $$\lambda_1\leq k$$ is introduced. Its definition is analogous to the combinatorial construction of the Hall-Littlewood polynomials. The $$A_{\lambda}^{(k)}[X;t]$$ are linearly independent and can be expanded in terms of Schur functions as $A_{\lambda}^{(k)}[X;t]=S_{\lambda}[X]+\sum_{\mu>\lambda} v_{\mu\lambda}^{(k)}(t) S_{\mu}[X],$ where the $$v_{\mu\lambda}^{(k)}(t)$$ are polynomials in $$t$$ with nonnegative coefficients. Furthermore, $$A_{\lambda}^{(k)}[X;t]=S_{\lambda}[X]$$ when $$k\geq \lambda_1 +\ell(\lambda)-1\leq |\lambda|$$.
A large number of very intriguing conjectures are made about the polynomials $$A_{\lambda}^{(k)}[X;t]$$. For example, they are conjectured to form a basis of $$V_k$$, the space of symmetric functions spanned by the modified Schur functions $$S_{\lambda}[X/(1-t)]$$ for $$\lambda_1\leq k$$. The most important conjecture about the new polynomials is however that $$H_{\mu}[X;q,t]$$ — the modified integral form of the Macdonald polynomial $$J_{\mu}[X;q,t]$$, $$H_{\mu}[X;q,t]=J_{\mu}[X/(1-t);q,t]$$ — expands positively in terms of the $$A_{\lambda}^{(k)}[X;t]$$: $H_{\mu}[X;q,t]=\sum_{\lambda} K_{\lambda\mu}^{(k)}(q,t) A^{(k)}_{\lambda}[X;t],$ with $$K_{\lambda\mu}^{(k)}(q,t)$$ a polynomial with nonnegative coefficients such that also $$K_{\lambda\mu}(q,t)-K_{\lambda\mu}^{(k)}(q,t)$$ has nonnegative coefficients. Here $$K_{\lambda\mu}(q,t)$$ are the $$(q,t)$$-Kostka coefficients defined by the above equation for $$k$$ sufficiently large: $H_{\mu}[X;q,t]=\sum_{\lambda} K_{\lambda\mu}(q,t) S_{\lambda}[X].$ This conjecture can be viewed as a strong version of the Macdonald positivity (ex-)conjecture since for $$k\geq |\mu|$$ it follows that $K_{\lambda\mu}^{(k)}(q,t)=K_{\lambda\mu}(q,t).$

##### MSC:
 500000 Symmetric functions and generalizations
##### Keywords:
symmetric functions
ACE
Full Text:
##### References:
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