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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:
05E05 Symmetric functions and generalizations
Software:
ACE
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References:
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