The transition matrix between the Specht and web bases is unipotent with additional vanishing entries
The transition matrix between the Specht and web bases is unipotent with additional vanishing entries
复制标题
Specht 和 Web 基础之间的转换矩阵是单能的，具有额外的消失条目
DOI：
10.1093/imrn/rnx164
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Julianna Tymoczko
中科院分区：
文献类型：
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作者：
Heather M. Russell;Julianna Tymoczko
我们比较了对称群的不可约表示的两个重要基：网基和Speht基。网络基础源于Temperley-Lieb代数和纽结理论的考虑。Speht基是对称群表示的经典代数和组合构造，它是在这种情况下通过称为Springer纤维的簇的几何而产生的。我们描述了一个图，它封装了这些基之间的组合关系，证明了存在唯一的方法(直到比例)将specht基映射到网络表示中，并利用这一方法恢复了Garsia-McLarnan的结果，即specht和web基之间的转移矩阵是上三角形的，沿对角线。然后，我们加强了他们的结果，以证明除非满足Web上的嵌套条件，否则某些额外的条目将消失。事实上，我们猜想，当网络图中存在某些有向路径时，转移矩阵的项是非负的并且是非零的。
We compare two important bases of an irreducible representation of the symmetric group: the web basis and the Specht basis. The web basis has its roots in the Temperley-Lieb algebra and knot-theoretic considerations. The Specht basis is a classic algebraic and combinatorial construction of symmetric group representations which arises in this context through the geometry of varieties called Springer fibers. We describe a graph that encapsulates combinatorial relations between each of these bases, prove that there is a unique way (up to scaling) to map the Specht basis into the web representation, and use this to recover a result of Garsia-McLarnan that the transition matrix between the Specht and web bases is upper-triangular with ones along the diagonal. We then strengthen their result to prove vanishing of certain additional entries unless a nesting condition on webs is satisfied. In fact we conjecture that the entries of the transition matrix are nonnegative and are nonzero precisely when certain directed paths exist in the web graph.