We study an action of the plactic algebra on bosonic particle configurations. These particle configurations together with the action of the plactic generators can be identified with crystals of the quantum analogue of the symmetric tensor representations of the special linear Lie algebra \(\mathfrak {s} \mathfrak {l}_{N}\). It turns out that this action factors through a quotient algebra that we call partic algebra, whose induced action on bosonic particle configurations is faithful. We describe a basis of the partic algebra explicitly in terms of a normal form for monomials, and we compute the center of the partic algebra.

中文翻译：

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Bosonic粒子构型的实用代数作用

我们研究了拟代数对玻子粒子构型的作用。这些粒子构型以及实际生成器的作用可以通过特殊线性李代数\（\ mathfrak {s} \ mathfrak {l} _ {N} \）的对称张量表示的量子模拟的晶体来识别。事实证明，这种作用是通过商代数（我们称为粒子代数）分解的，它对玻色子粒子构型的诱导作用是忠实的。我们以单项式的标准形式明确描述了分式代数的基础，并计算了分式代数的中心。