Our aim is to alternate elements of the general theory with significant examples. We deal now with synmietric functions. In this chapter we will develop some of the very basic theorems on symmetric functions, in part as a way to give a look into 19* century invariant theory, but as well to establish some useful formulas which will show their full meaning only after developing the representation theory of the linear and symmetric groups.

On symmetric functions.

Kohnert polynomials are polynomials indexed by unit cell diagrams in the first quadrant defined earlier by the author and Searles that give a common generalization of Schubert polynomials and Demazure characters for the general linear group. Demazure crystals are certain truncations of normal crystals whose characters are Demazure characters. For each diagram satisfying a southwest condition, we construct a Demazure crystal whose character is the Kohnert polynomial for the given diagram, resolving an earlier conjecture of the author and Searles that these polynomials expand nonnegatively into Demazure characters. We give explicit formulas for the expansions with applications including a characterization of those diagrams for which the corresponding Kohnert polynomial is a single Demazure character.

Demazure crystals for Kohnert polynomials

The product monomial crystal was defined by Kamnitzer, Tingley, Webster, Weekes, and Yacobi for any semisimple simply-laced Lie algebra $\mathfrak{g}$, and depends on a collection of parameters $\mathbf{R}$. We show that a family of truncations of this crystal are Demazure crystals, and give a Demazure-type formula for the character of each truncation, and the crystal itself. This character formula shows that the product monomial crystal is the crystal of a generalised Demazure module, as defined by Lakshmibai, Littelmann and Magyar. In type $A$, we show the product monomial crystal is the crystal of a generalised Schur module associated to a column-convex diagram depending on $\mathbf{R}$.

/pdf/a-demazure-character-formula-for-the-product-monomial-pbyd9xzilv.pdf

A Demazure character formula for the product monomial crystal

Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include $k$-Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of $U_q(\widehat{\mathfrak{sl}}_\ell)$-generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals.

pdf/demazure-crystals-and-the-schur-positivity-of-catalan-46cqsm30kp.pdf

Demazure crystals and the Schur positivity of Catalan functions

In this note, we provide a short proof of Theorem 3.3 in the paper titled \emph{Crystals, semistandard tableaux and cyclic sieving phenomenon}, by Y.-T.~Oh and E.~Park, which concerns a cyclic sieving phenomenon on semi-standard Young tableaux. We also extend the result to skew shapes.

pdf/free-action-and-cyclic-sieving-on-skew-semi-standard-young-3z1f27cgu4.pdf

Free action and cyclic sieving on skew semi-standard Young tableaux.

We give an explicit, nonnegative formula for the expansion of nonsymmetric Macdonald polynomials specialized at $t=0$ in terms of Demazure characters. Our formula results from constructing Demazure crystals whose characters are the nonsymmetric Macdonald polynomials, which also gives a new proof that these specialized nonsymmetric Macdonald polynomials are positive graded sums of Demazure characters. Demazure crystals are certain truncations of classical crystals that give a combinatorial skeleton for Demazure modules. To prove our construction, we develop further properties of Demazure crystals, including an efficient algorithm for computing their characters from highest weight elements. As a corollary, we obtain a new formula for the Schur expansion of Hall--Littlewood polynomials in terms of a simple statistic on highest weight elements of our crystals.

Demazure crystals for specialized nonsymmetric Macdonald polynomials

/pdf/affine-demazure-crystals-for-specialized-nonsymmetric-9cilc8esw3.pdf

Affine Demazure crystals for specialized nonsymmetric Macdonald polynomials

We prove a conjecture of Cautis and Sussan providing a categorification of the Boson-Fermion correspondence as formulated by Frenkel and Kac. We lift the Bernstein operators to infinite chain complexes in Khovanov's Heisenberg category H and from them construct categorical analogues of the Kac-Frenkel fermionic vertex operators. These fermionic functors are then shown to satisfy categorical Clifford algebra relations, solving a conjecture of Cautis and Sussan. We also prove another conjecture of Cautis and Sussan demonstrating that the categorical Fock space representation of H is a direct summand of the regular representation by showing that certain infinite chain complexes are categorical Fock space idempotents. In the process, we enhance the graphical calculus of H by lifting various Littlewood-Richardson branching isomorphisms to the Karoubian envelope of H.

Categorical Bernstein Operators and the Boson-Fermion Correspondence

We prove a conjecture of Cautis and Sussan providing a categorification of the Boson-Fermion correspondence as formulated by Frenkel and Kac. We lift the Bernstein operators to infinite chain complexes in Khovanov’s Heisenberg category 
$${\mathcal {H}}$$

 and from them construct categorical analogues of the Kac-Frenkel fermionic vertex operators. These fermionic functors are then shown to satisfy categorical Clifford algebra relations, solving a conjecture of Cautis and Sussan. We also prove another conjecture of Cautis and Sussan demonstrating that the categorical Fock space representation of 
$${\mathcal {H}}$$

 is a direct summand of the regular representation by showing that certain infinite chain complexes are categorical Fock space idempotents. In the process, we enhance the graphical calculus of 
$${\mathcal {H}}$$

 by lifting various Littlewood-Richardson branching isomorphisms to the Karoubian envelope of 
$${\mathcal {H}}$$

.

Categorical Bernstein operators and the Boson-Fermion correspondence

pdf/demazure-crystals-for-specialized-nonsymmetric-macdonald-1vstqm10v5.pdf

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Demazure crystals for specialized nonsymmetric Macdonald polynomials

Affine Demazure crystals for specialized nonsymmetric Macdonald polynomials

Categorical Bernstein Operators and the Boson-Fermion Correspondence

Categorical Bernstein operators and the Boson-Fermion correspondence

Demazure crystals for specialized nonsymmetric Macdonald polynomials