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We establish an explicit embedding of a quantum affine sl_n into a quantum affine sl_{n+1} . This embedding serves as a common generalization of two natural, but seemingly unrelated embeddings, one on the quantum affine Schur algebra level and the other on the non-quantum level. The embedding on the quantum affine Schur algebras is used extensively in the analysis of canonical bases of quantum affine sl_n and gl_n. The embedding on the non-quantum level is used crucially in a work of Riche and Williamson on the study of modular representation theory of general linear groups over a finite field. The same embedding is also used in a work of Maksimau on the categorical representations of affine general linear algebras. We further provide a more natural compatibility statement of the em- bedding on the idempotent version with that on the quantum affine Schur algebra level. A gl_n-variant of the embedding is also established. 

Abstract We show the positivity of the canonical basis for a modified quantum affine $$\mathfrak{sl}_n$$ under the comultiplication. Moreover, we establish the positivity of the i-canonical basis in [21] with respect to the coideal subalgebra structure. 

We study fixed-point loci of Nakajima varieties under symplectomorphisms and their antisymplectic cousins, which are compositions of a diagram isomorphism, a reflection functor, and a transpose defined by certain bilinear forms. These subvarieties provide a natural home for geometric representation theory of symmetric pairs. In particular, the cohomology of a Steinberg-type variety of the symplectic fixed-point subvarieties is conjecturally related to the universal enveloping algebra of the subalgebra in a symmetric pair. The latter symplectic subvarieties are further used to geometrically construct an action of a twisted Yangian on a torus equivariant cohomology of Nakajima varieties. In the type A case, these subvarieties provide a quiver model for partial Springer resolutions of nilpotent Slodowy slices of classical groups and associated symmetric spaces, which leads to a rectangular symmetry and a refinement of Kraft-Procesi row/column removal reductions.