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In [Math. Ann. 367 (2017), pp. 1517–1586] Bar-Natan and the first author show that solutions to the Kashiwara–Vergne equations are in bijection with certain knot invariants: homomorphic expansions of welded foams. Welded foams are a class of knotted tubes in \mathbb {R}^4, which can be finitely presented algebraically as a circuit algebra , or equivalently, a wheeled prop. In this paper we describe the Kashiwara-Vergne groups \mathsf {KV} and \mathsf {KRV}—the symmetry groups of Kashiwara-Vergne solutions—as automorphisms of the completed circuit algebras of welded foams, and their associated graded circuit algebras of arrow diagrams, respectively. Finally, we provide a description of the graded Grothendieck-Teichmüller group \mathsf {GRT}_1 as automorphisms of arrow diagrams. [ABSTRACT FROM AUTHOR]

Starting from an integrable rank-n vertex model, we construct an explicit family of partition functions indexed by compositions \mu = (\mu _1,\dots,\mu _n). Using the Yang–Baxter algebra of the model and a certain rotation operation that acts on our partition functions, we show that they are eigenfunctions of the Cherednik–Dunkl operators Y_i for all 1 \leqslant i \leqslant n, and are thus equal to nonsymmetric Macdonald polynomials E_{\mu }. Our partition functions have the combinatorial interpretation of ensembles of coloured lattice paths which traverse a cylinder. Applying a simple bijection to such path ensembles, we show how to recover the well-known combinatorial formula for E_{\mu } due to Haglund–Haiman–Loehr. [ABSTRACT FROM AUTHOR]

We introduce a family of ideals In,λ,s in Q[x1, . . . , xn] for λ a partition of k ≤ n and an integer s ≥ l (λ). This family contains both the Tanisaki ideals Iλ and the ideals In,k of Haglund-Rhoades-Shimozono as special cases. We study the corresponding quotient rings Rn,λ,s as symmetric group modules. When n = k and s is arbitrary, we recover the Garsia-Procesi modules, and when λ = (1k) and s = k, we recover the generalized coinvariant algebras of Haglund-Rhoades-Shimozono. We give a monomial basis for Rn,λ, s in terms of (n,λ,s)-staircases, unifying the monomial bases studied by Garsia-Procesi and Haglund-Rhoades-Shimozono. We realize the Sn-module structure of Rn,λ,s in terms of an action on (n, λ, s)-ordered set partitions. We find a formula for the Hilbert series of Rn,λ,s in terms of inversion and diagonal inversion statistics on a set of fillings in bijection with (n, λ, s)-ordered set partitions. Furthermore, we prove an expansion of the graded Frobenius characteristic of our rings into Gessel's fundamental quasisymmetric basis. We connect our work with Eisenbud-Saltman rank varieties using results of Weyman. As an application of our results on Rn,λ,s, we give a monomial basis, Hilbert series formula, and graded Frobenius characteristic formula for the coordinate ring of the scheme-theoretic intersection of a rank variety with diagonal matrices. [ABSTRACT FROM AUTHOR]

Assume that D is a Krull-Schmidt, Hom-finite triangulated category with a Serre functor and a cluster-tilting object T. We introduce the notion of relative cluster tilting objects, and T[1]-cluster tilting objects in D, which are a generalization of cluster-tilting objects. When D is 2-Calabi-Yau, the relative cluster tilting objects are cluster-tilting. Let Λ = EndDop(T) be the opposite algebra of the endomorphism algebra of T. We show that there exists a bijection between T[1]-cluster tilting objects in D and support τ-tilting Λ- modules, which generalizes a result of Adachi--Iyama--Reiten [τ-tilting theory, Compos. Math. 150 (2014), no. 3, 415-452]. We develop a basic theory on T[1]-cluster tilting objects. In particular, we introduce a partial order on the set of T[1]-cluster tilting objects and mutation of T[1]-cluster tilting objects, which can be regarded as a generalization of "cluster-tilting mutation". As an application, we give a partial answer to a question posed in Adachi--Iyama--Reiten, loc. cit. [ABSTRACT FROM AUTHOR]