布尔格和子空间格的Segre幂的同调
Homology of Segre powers of Boolean and subspace lattices
AI总结：
研究布尔格和子空间格的Segre幂的同调表示，揭示对称群作用下的分解公式及秩选同调的不变量关系。
摘要：
Segre poset积由Björner和Welker（2005）定义。我们研究了布尔格B_n的t重Segre幂B_n^{(t)}的同调表示。对称群sym_n的直接积sym_n^{\times t}作用于B_n^{(t)}的秩选子poset的同调上。我们给出了整个poset同调分解成sym_n^{\times t}-不可约表示的显式公式，以及对称群sym_n的对角作用的公式。对于秩选同调，我们证明了产品Frobenius特征式的稳定主专化与t重Segre幂的子空间格的相应秩选不变量相等。
英文摘要：
Segre products of posets were defined by Björner and Welker (2005). We investigate the homology representations of the $t$-fold Segre power $B_n^{(t)}$ of the Boolean lattice $B_n$. The direct product $\sym_n^{\times t}$ of the symmetric group $\sym_n$ acts on the homology of rank-selected subposets of $B_n^{(t)}$. We give an explicit formula for the decomposition into $\sym_n^{\times t}$-irreducibles of the homology of the full poset, as well as formulas for the diagonal action of the symmetric group $\sym_n$. For the rank-selected homology, we show that the stable principal specialisation of the product Frobenius characteristic of the $\sym_n^{\times t}$-module coincides with the corresponding rank-selected invariant of the $t$-fold Segre power of the subspace lattice.
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