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Leclerc's conjecture on a cluster structure for type A Richardson varieties

. Leclerc [Lec16] constructed a conjectural cluster structure on Richardson varieties in simply laced types using cluster categories. We show that in type A, his conjectural cluster structure is in fact a cluster structure. We do this by comparing Leclerc’s construction with another cluster structure on type A Richardson varieties due to Ingermanson [Ing19]. Ingermanson’s construction uses the combinatorics of wiring diagrams and the Deodhar stratiﬁcation. Though the two cluster structures are deﬁned very diﬀerently, we show that the quivers coincide and clusters are related by the twist map for Richardson varieties, recently deﬁned by Galashin–Lam [GL22].