Weak orders on symmetric groups and posets of support $τ$-tilting modules

We give a necessary and sufficient condition for that the support $τ$-tilting poset of a finite dimensional algebra is isomorphic to the poset of symmetric group with weak order. Moreover we show that there are infinitely many finite dimensional algebras whose support $τ$-tilting posets are isomorphic to the poset of symmetric group with weak order.

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