对称 Fibonacci 分配格与特殊线性李代数的表示
Symmetric Fibonaccian distributive lattices and representations of the special linear Lie algebras
AI总结：
本文构造一族秩对称、菱形着色的 Fibonacci 分配格，统一已有格结构，并用其自然实现特殊线性李代数表示及具有极值性质的权基。
摘要：
我们提出一族秩对称、菱形着色的分配格，它们与 Fibonacci 序列及其若干推广具有自然联系。这些格重新解释并统一了文献中以不同方式出现的一些未着色或以不同方式着色的格的描述。我们证明，我们的对称 Fibonacci 格自然实现特殊线性李代数的某些（通常是可约的）表示，其中权基向量被实现为格元素，而李代数生成元沿每个格的覆盖有向图边作用。我们给出证据，表明每个这样的权基都具有某些显著的极值性质。我们提供了格基数和秩生成函数的新描述，并提出若干猜想与开放问题。全文始终与 OEIS 中的整数序列建立联系。
英文摘要：
We present a family of rank symmetric diamond-colored distributive lattices that are naturally related to the Fibonacci sequence and certain of its generalizations. These lattices re-interpret and unify descriptions of some un- or differently-colored lattices found variously in the literature. We demonstrate that our symmetric Fibonaccian lattices naturally realize certain (often reducible) representations of the special linear Lie algebras, with weight basis vectors realized as lattice elements and Lie algebra generators acting along the covering digraph edges of each lattice. We present evidence that each such weight basis possesses certain distinctive extremal properties. We provide new descriptions of the lattice cardinalities and rank generating functions and offer several conjectures/open problems. Throughout, we make connections with integer sequences from the OEIS.
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