Noncommutative Schur Functions Research Articles

Dual graphs from noncommutative and quasisymmetric Schur functions

By establishing relations between operators on compositions, we show that the posets of compositions arising from the right and left Pieri rules for noncommutative Schur functions can each be endowed with both the structure of dual graded graphs and dual filtered graphs when paired with the poset of compositions arising from the Pieri rules for quasisymmetric Schur functions and its deformation.

Quasisymmetric and noncommutative skew Pieri rules

In this note we derive skew Pieri rules in the spirit of Assaf–McNamara for skew quasisymmetric Schur functions using the Hopf algebraic techniques of Lam–Lauve–Sottile, and recover the original rules of Assaf–McNamara as a special case. We then apply these techniques a second time to obtain skew Pieri rules for skew noncommutative Schur functions.

Kronecker coefficients and noncommutative super Schur functions

The theory of noncommutative Schur functions can be used to obtain positive combinatorial formulae for the Schur expansion of various classes of symmetric functions, as shown by Fomin and Greene [11]. We develop a theory of noncommutative super Schur functions and use it to prove a positive combinatorial rule for the Kronecker coefficients gλμν where one of the partitions is a hook, recovering previous results of the two authors [7,22]. This method also gives a precise connection between this rule and a heuristic for Kronecker coefficients first investigated by Lascoux [19].

Dual immaculate quasisymmetric functions expand positively into Young quasisymmetric Schur functions

We describe a combinatorial formula for the coefficients when the dual immaculate quasisymmetric functions are decomposed into Young quasisymmetric Schur functions. We prove this using an analogue of Schensted insertion. Using this result, we give necessary and sufficient conditions for a dual immaculate quasisymmetric function to be symmetric. Moreover, we show that the product of a Schur function and a dual immaculate quasisymmetric function expands positively in the Young quasisymmetric Schur basis. We also discuss the decomposition of the Young noncommutative Schur functions into the immaculate functions. Finally, we provide a Remmel–Whitney-style rule to generate the coefficients of the decomposition of the dual immaculates into the Young quasisymmetric Schurs algorithmically and an analogous rule for the decomposition of the dual bases.

Noncommutative Schur functions, switchboards, and Schur positivity

We review and further develop a general approach to Schur positivity of symmetric functions based on the machinery of noncommutative Schur functions. This approach unifies ideas of Assaf, Lam, and Greene and the second author.

A Murnaghan–Nakayama rule for noncommutative Schur functions

We prove a Murnaghan–Nakayama rule for noncommutative Schur functions introduced by Bessenrodt, Luoto and van Willigenburg. In other words, we give an explicit combinatorial formula for expanding the product of a noncommutative power sum symmetric function and a noncommutative Schur function in terms of noncommutative Schur functions. In direct analogy to the classical Murnaghan–Nakayama rule, the summands are computed using a noncommutative analogue of border strips, and have coefficients ±1 determined by the height of these border strips. The rule is proved by interpreting the noncommutative Pieri rules for noncommutative Schur functions in terms of box-adding operators on compositions.

What makes a $$\mathbf{D}_0$$ D 0 graph Schur positive?

We define a $${D}_0$$D0graph to be a graph whose vertex set is a subset of permutations of n, with edges of the form $$\cdots \mathsf {bac} \cdots \leftrightsquigarrow \cdots \mathsf {bca} \cdots $$źbacźźźbcaź or $$\cdots \mathsf {acb} \cdots \leftrightsquigarrow \cdots \mathsf {cab} \cdots $$źacbźźźcabź (Knuth transformations), or $$\cdots \mathsf {bac} \cdots \leftrightsquigarrow \cdots \mathsf {acb} \cdots $$źbacźźźacbź or $$\cdots \mathsf {bca} \cdots \leftrightsquigarrow \cdots \mathsf {cab} \cdots $$źbcaźźźcabź (rotation transformations), such that whenever the Knuth and rotation transformations at positions $$i-1, i, i+1$$i-1,i,i+1 are available at a vertex, exactly one of these is an edge. The generating function of such a graph is the sum of the quasisymmetric functions associated to the descent sets of its vertices. Assaf studied D$$_0$$0 graphs in (Dual equivalence and Schur positivity, http://www-bcf.usc.edu/~shassaf/degs.pdf, 2014) and showed that they provide a rich source of examples of the D graphs of (Dual equivalence graphs and a combinatorial proof of LLT and Macdonald positivity, http://www-bcf.usc.edu/~shassaf/positivity.pdf, 2014). A key construction of Assaf expresses the coefficient of $$q^t$$qt in an LLT polynomial as the generating function of a certain D$$_0$$0 graph. LLT polynomials are known to be Schur positive by work of Grojnowski-Haiman, and experimentation shows that many D$$_0$$0 graphs have Schur positive generating functions, which suggests a vast generalization of LLT positivity in this setting. As part of a series of papers, we study D$$_0$$0 graphs using the Fomin-Greene theory of noncommutative Schur functions. We construct a D$$_0$$0 graph whose generating function is not Schur positive by solving a linear program related to a certain noncommutative Schur function. We go on to construct a D graph on the same vertex set as this D$$_0$$0 graph.

Noncommutative irreducible characters of the symmetric group and noncommutative Schur functions

In the Hopf algebra of symmetric functions, Sym, the basis of Schur functions is distinguished since every Schur function is isomorphic to an irreducible character of a symmetric group under the Frobenius characteristic map. In this note we show that in the Hopf algebra of noncommutative symmetric functions, NSym, of which Sym is a quotient, the recently discovered basis of noncommutative Schur functions exhibits that every noncommutative Schur function is isomorphic to a noncommutative irreducible character of a symmetric group when working in noncommutative character theory. We simultaneously show that a second basis of NSym consisting of Young noncommutative Schur functions also satises that every element is isomorphic to a noncommutative irreducible character of a symmetric group.

Skew quasisymmetric Schur functions and noncommutative Schur functions

Recently a new basis for the Hopf algebra of quasisymmetric functions QSym, called quasisymmetric Schur functions, has been introduced by Haglund, Luoto, Mason, van Willigenburg. In this paper we extend the definition of quasisymmetric Schur functions to introduce skew quasisymmetric Schur functions. These functions include both classical skew Schur functions and quasisymmetric Schur functions as examples, and give rise to a new poset LC that is analogous to Young's lattice. We also introduce a new basis for the Hopf algebra of noncommutative symmetric functions NSym. This basis of NSym is dual to the basis of quasisymmetric Schur functions and its elements are the pre-image of the Schur functions under the forgetful map χ:NSym→Sym. We prove that the multiplicative structure constants of the noncommutative Schur functions, equivalently the coefficients of the skew quasisymmetric Schur functions when expanded in the quasisymmetric Schur basis, are nonnegative integers, satisfying a Littlewood–Richardson rule analogue that reduces to the classical Littlewood–Richardson rule under χ.As an application we show that the morphism of algebras from the algebra of Poirier–Reutenauer to Sym factors through NSym. We also extend the definition of Schur functions in noncommuting variables of Rosas–Sagan in the algebra NCSym to define quasisymmetric Schur functions in the algebra NCQSym. We prove these latter functions refine the former and their properties, and project onto quasisymmetric Schur functions under the forgetful map. Lastly, we show that by suitably labeling LC, skew quasisymmetric Schur functions arise in the theory of Pieri operators on posets.

The shifted plactic monoid

We introduce a shifted analog of the plactic monoid of Lascoux and Schutzenberger, the shifted plactic monoid. It can be defined in two different ways: via the shifted Knuth relations, or using Haiman’s mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood–Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur P-function; a shifted counterpart of the Lascoux–Schutzenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.

The shifted plactic monoid (extended abstract)

We introduce a shifted analog of the plactic monoid of Lascoux and Schützenberger, the \emphshifted plactic monoid. It can be defined in two different ways: via the \emphshifted Knuth relations, or using Haiman's mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood-Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur P-function; a shifted counterpart of the Lascoux-Schützenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.

Ribbon Schur operators

A new combinatorial approach to the ribbon tableaux generating functions and q -Littlewood–Richardson coefficients of Lascoux et al. [A. Lascoux, B. Leclerc, J.-Y. Thibon, Ribbon tableaux, Hall–Littlewood symmetric functions, quantum affine algebras, and unipotent varieties, J. Math. Phys. 38 (3) (1997) 1041–1068] is suggested. We define operators which add ribbons to partitions and following Fomin and Greene [S. Fomin, C. Greene, Noncommutative Schur functions and their applications, Discrete Math. 193 (1998) 179–200] study non-commutative symmetric functions in these operators. This allows us to give combinatorial interpretations for some (skew) q -Littlewood–Richardson coefficients whose non-negativity appears not to be known. Our set-up also leads to a new proof of the action of the Heisenberg algebra on the Fock space of U q ( s l ̂ n ) due to Kashiwara et al. [M. Kashiwara, T. Miwa, E. Stern, Decomposition of q -deformed Fock spaces, Selecta Math. 1 (1996) 787–805].

Noncommutative Schur functions and their applications

We develop a theory of Schur functions in noncommuting variables, assuming commutation relations that are satisfied in many well-known associative algebras. As an application of our theory, we prove Schur-positivity and obtain generalized Littlewood–Richardson and Murnaghan–Nakayama rules for a large class of symmetric functions, including stable Schubert and Grothendieck polynomials.

Noncommutative Schubert Calculus and Grothendieck Polynomials

In this paper we extend the work of Fomin and Greene on noncommutative Schur functions by defining noncommutative analogs of Schubert polynomials. If the variables satisfy certain relations (essentially the same as those needed in the theory of noncommutative Schur functions), we prove a Pieri-type formula and a Cauchy identity for our noncommutative polynomials. Our results imply the conjecture of Fomin and Kirillov concerning the expansion of an arbitrary Grothendieck polynomial in the basis of Schubert polynomials; we also present a combinatorial interpretation for the coefficients of the expansion. We conclude with some open problems related to it.

Noncommutative schur functions and their applications

We develop a theory of Schur functions in noncommuting variables, assuming commutation relations that are satisfied in many well-known associative algebras. As an application of our theory, we prove Schur-positivity and obtain generalized Littlewood-Richardson and Murnaghan-Nakayama rules for a large class of symmetric functions, including stable Schubert and Grothendieck polynomials.