Monoidal categories associated with strata of flag manifolds

다국어 초록 (Multilingual Abstract)

<P><B>Abstract</B></P> <P>We construct a monoidal category <SUB> C w , v </SUB> which categorifies the doubly-invariant algebra C <SUP> N ′ </SUP> ( w ) <SUP> [ N ] N ( v ) </SUP> associated with Weyl group elements <I>w</I> and <I>v</I>. It gives, after a localization, the coordinate algebra C [ <SUB> R w , v </SUB> ] of the open Richardson variety associated with <I>w</I> and <I>v</I>. The category <SUB> C w , v </SUB> is realized as a subcategory of the graded module category of a quiver Hecke algebra <I>R</I>. When v = id , <SUB> C w , v </SUB> is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent coordinate algebra <SUB> A q </SUB> <SUB> ( n ( w ) ) Z [ q , <SUP> q − 1 </SUP> ] </SUB> given by Kang–Kashiwara–Kim–Oh. We show that the category <SUB> C w , v </SUB> contains special determinantial modules M ( <SUB> w ≤ k </SUB> Λ , <SUB> v ≤ k </SUB> Λ ) for k = 1 , … , ℓ ( w ) , which commute with each other. When the quiver Hecke algebra <I>R</I> is symmetric, we find a formula of the degree of <I>R</I>-matrices between the determinantial modules M ( <SUB> w ≤ k </SUB> Λ , <SUB> v ≤ k </SUB> Λ ) . When it is of finite <I>ADE</I> type, we further prove that there is an equivalence of categories between <SUB> C w , v </SUB> and <SUB> C u </SUB> for w , u , v ∈ W with w = v u and ℓ ( w ) = ℓ ( v ) + ℓ ( u ) .</P>