Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Shi arrangements and low elements in affine Coxeter groups   Given an affine Coxeter group $W$, the corresponding Shi arrangement is a
refinement of the corresponding Coxeter hyperplane arrangements that was
introduced by Shi to study Kazhdan-Lusztig cells for $W$. In particular, Shi
showed that each region of the Shi arrangement contains exactly one element of
minimal length in $W$. Low elements in $W$ were introduced to study the word
problem of the corresponding Artin-Tits (braid) group and turns out to produce
automata to study the combinatorics of reduced words in $W$.
  In this article, we show in the case of an affine Coxeter group that the set
of minimal length elements of the regions in the Shi arrangement is precisely
the set of low elements, settling a conjecture of Dyer and the second author in
this case. As a byproduct of our proof, we show that the descent-walls -- the
walls that separate a region from the fundamental alcove -- of any region in
the Shi arrangement are precisely the descent walls of the alcove of its
corresponding low element.

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The categories of this paper are: quant-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: stat
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: supr-con
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The categories of this paper are: q-alg
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The categories of this paper are: q-fin
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: adap-org
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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex