论文标题
$ M $ - 符合功能，非对称麦克唐纳多项式和积极性猜想
$m$-Symmetric functions, non-symmetric Macdonald polynomials and positivity conjectures
论文作者
论文摘要
我们研究了$ m $ $ $ $ $ $的空间，由多项式组成，这些功能由多项式组成，这些函数在变量中是对称的$ x_ {m+1}，x_ {m+2}，x_ {m+3}，\ dots $，但在变量中没有特殊对称性，但在变量中$ x_1 $ x_1，\ dots，\ dots，x____m。我们通过$ t $ - 对称非对称的麦克唐纳多项式的多项式获得$ M $ $ $ - $ $ MACDONALD的多项式，并表明它们形成了$ r_m $的基础。我们通过涉及其双重基础，Tableaux Combinatorics和Hecke代数发生器的一个复杂的过程来定义$ m $ $ $ s的Schur函数，然后证明其一些最基本的属性。我们猜想，$ M $ symmetric MacDonald多项式（适当地归一化且众所周知地修改）以$ M $ -SMMETRIC SCHUR功能的积极扩展。我们在$（q，t）$ - koska系数上获得关系$ k_ {ωλ}（q，t）$中的$ m $ - $ - 对称世界，并特别表明通常的$（q，q，t）$ - koska系数是$ k_ {ωλ}（q om q，q，q，q，t）的特殊情况。最后，我们表明，当$ m $很大时，就非对称的麦克唐纳多条件的扩展而言，就非对称的Hall-Littlewood多项式而言，阳性猜想（Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo a Modulo。
We study the space, $R_m$, of $m$-symmetric functions consisting of polynomials that are symmetric in the variables $x_{m+1},x_{m+2},x_{m+3},\dots$ but have no special symmetry in the variables $x_1,\dots,x_m$. We obtain $m$-symmetric Macdonald polynomials by $t$-symmetrizing non-symmetric Macdonald polynomials, and show that they form a basis of $R_m$. We define $m$-symmetric Schur functions through a somewhat complicated process involving their dual basis, tableaux combinatorics, and the Hecke algebra generators, and then prove some of their most elementary properties. We conjecture that the $m$-symmetric Macdonald polynomials (suitably normalized and plethystically modified) expand positively in terms of $m$-symmetric Schur functions. We obtain relations on the $(q,t)$-Koska coefficients $K_{ΩΛ}(q,t)$ in the $m$-symmetric world, and show in particular that the usual $(q,t)$-Koska coefficients are special cases of the $K_{ΩΛ}(q,t)$'s. Finally, we show that when $m$ is large, the positivity conjecture, modulo a certain subspace, becomes a positivity conjecture on the expansion of non-symmetric Macdonald polynomials in terms of non-symmetric Hall-Littlewood polynomials.
