Cylindric skew Schur functions

Cylindric skew Schur functions, which are a generalisation of skew Schur functions, arise naturally in the study of P-partitions. Also, recent work of A. Postnikov shows they have a strong connection with a problem of considerable current interest: that of finding a combinatorial proof of the non-negativity of the 3-point Gromov-Witten invariants. After explaining these motivations, we study cylindric skew Schur functions from the point of view of Schur-positivity. Using a result of I. Gessel and C. Krattenthaler, we generalise a formula of A. Bertram, I. Ciocan-Fontanine and W. Fulton, thus giving an expansion of an arbitrary cylindric skew Schur function in terms of skew Schur functions. While we show that no non-trivial cylindric skew Schur functions are Schur-positive, we conjecture that this can be reconciled using the new concept of cylindric Schur-positivity.Comment: 32 pages, 14 figures. Minor expository improvements. Version to appear in Advances in Mathematic

The projective cover of tableau-cyclic indecomposable Hn(0)H_n(0)-modules

Let $\alpha$ be a composition of $n$ and $\sigma$ a permutation in $\mathfrak{S}_{\ell(\alpha)}$. This paper concerns the projective covers of $H_n(0)$-modules $\mathcal{V}_\alpha$, $X_\alpha$ and $\mathbf{S}^\sigma_{\alpha}$, which categorify the dual immaculate quasisymmetric function, the extended Schur function, and the quasisymmetric Schur function when $\sigma$ is the identity, respectively. First, we show that the projective cover of $\mathcal{V}_\alpha$ is the projective indecomposable module $\mathbf{P}_\alpha$ due to Norton, and $X_\alpha$ and the $\phi$-twist of the canonical submodule $\mathbf{S}^{\sigma}_{\beta,C}$ of $\mathbf{S}^\sigma_{\beta}$ for $(\beta,\sigma)$'s satisfying suitable conditions appear as $H_n(0)$-homomorphic images of $\mathcal{V}_\alpha$. Second, we introduce a combinatorial model for the $\phi$-twist of $\mathbf{S}^\sigma_{\alpha}$ and derive a series of surjections starting from $\mathbf{P}_\alpha$ to the $\phi$-twist of $\mathbf{S}^{\mathrm{id}}_{\alpha,C}$. Finally, we construct the projective cover of every indecomposable direct summand $\mathbf{S}^\sigma_{\alpha, E}$ of $\mathbf{S}^\sigma_{\alpha}$. As a byproduct, we give a characterization of triples $(\sigma, \alpha, E)$ such that the projective cover of $\mathbf{S}^\sigma_{\alpha, E}$ is indecomposable.Comment: 41 page

kk-Schur functions and affine Schubert calculus

This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculusComment: 213 pages; conference website: http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/, updates and corrections since v1. This material is based upon work supported by the National Science Foundation under Grant No. DMS-065264

Knot invariants and higher representation theory

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl_2 and sl_3 and by Mazorchuk-Stroppel and Sussan for sl_n. Our technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is $\mathfrak{sl}_n$, we show that these categories agree with certain subcategories of parabolic category O for gl_k. We also investigate the finer structure of these categories: they are standardly stratified and satisfy a double centralizer property with respect to their self-dual modules. The standard modules of the stratification play an important role as test objects for functors, as Vermas do in more classical representation theory. The existence of these representations has consequences for the structure of previously studied categorifications. It allows us to prove the non-degeneracy of Khovanov and Lauda's 2-category (that its Hom spaces have the expected dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke algebras are symmetric Frobenius. In work of Reshetikhin and Turaev, the braiding and (co)evaluation maps between representations of quantum groups are used to define polynomial knot invariants. We show that the categorifications of tensor products are related by functors categorifying these maps, which allow the construction of bigraded knot homologies whose graded Euler characteristics are the original polynomial knot invariants.Comment: 99 pages. This is a significantly rewritten version of arXiv:1001.2020 and arXiv:1005.4559; both the exposition and proofs have been significantly improved. These earlier papers have been left up mainly in the interest of preserving references. v3: final version, to appear in Memoirs of the AMS. Proof of nondegeneracy moved to separate erratu