Schur Basis Research Articles

Extended Schur functions and bases related by involutions

We introduce two new bases of QSym, the reverse extended Schur functions and the row-strict reverse extended Schur functions, as well as their duals in NSym, the reverse shin functions and row-strict reverse shin functions. These bases are the images of the extended Schur basis and shin basis under the involutions ρ and ω, which generalize the classical involution ω on the symmetric functions. In addition, we prove a Jacobi-Trudi rule for certain shin functions using creation operators. We define skew extended Schur functions and skew-II extended Schur functions based on left and right actions of NSym and QSym respectively. We then use the involutions ρ and ω to translate these and other known results to our reverse and row-strict reverse bases.

Efficient classical algorithms for simulating symmetric quantum systems

In light of recently proposed quantum algorithms that incorporate symmetries in the hope of quantum advantage, we show that with symmetries that are restrictive enough, classical algorithms can efficiently emulate their quantum counterparts given certain classical descriptions of the input. Specifically, we give classical algorithms that calculate ground states and time-evolved expectation values for permutation-invariant Hamiltonians specified in the symmetrized Pauli basis with runtimes polynomial in the system size. We use tensor-network methods to transform symmetry-equivariant operators to the block-diagonal Schur basis that is of polynomial size, and then perform exact matrix multiplication or diagonalization in this basis. These methods are adaptable to a wide range of input and output states including those prescribed in the Schur basis, as matrix product states, or as arbitrary quantum states when given the power to apply low depth circuits and single qubit measurements.

Partial Symmetries of Iterated Plethysms

This work highlights the existence of partial symmetries in large families of iterated plethystic coefficients. The plethystic coefficients involved come from the expansion in the Schur basis of iterated plethysms of Schur functions indexed by one-row partitions.The partial symmetries are described in terms of an involution on partitions, the flip involution, that generalizes the ubiquitous $$\omega $$ involution. Schur-positive symmetric functions possessing this partial symmetry are termed flip-symmetric. The operation of taking plethysm with $$s_\lambda $$ preserves flip-symmetry, provided that $$\lambda $$ is a partition of two. Explicit formulas for the iterated plethysms $$s_2\circ s_b\circ s_a$$ and $$s_c\circ s_2\circ s_a$$ , with a, b, and c $$\ge $$ 2 allow us to show that these two families of iterated plethysms are flip-symmetric. The article concludes with some observations, remarks, and open questions on the unimodality and asymptotic normality of certain flip-symmetric sequences of iterated plethystic coefficients.

Petrie symmetric functions

For any positive integer $k$ and nonnegative integer $m$, we consider the symmetric $G\left( k,m\right)$ defined as the sum of all monomials of degree $m$ that involve only exponents smaller than $k$. We call $G\left( k,m\right)$ a symmetric function in honor of Flinders Petrie, as the coefficients in its expansion in the Schur basis are determinants of Petrie matrices (and thus belong to $\left\{ 0,1,-1\right\} $ by a classical result of Gordon and Wilkinson). More generally, we prove a Pieri-like rule for expanding a product of the form $G\left( k,m\right) \cdot s_{\mu}$ in the Schur basis whenever $\mu$ is a partition; all coefficients in this expansion belong to $\left\{ 0,1,-1\right\} $. We also show that $G\left( k,1\right) ,G\left( k,2\right) ,G\left( k,3\right) ,\ldots$ form an algebraically independent generating set for the symmetric functions when $1-k$ is invertible in the base ring, and we prove a conjecture of Liu and Polo about the expansion of $G\left( k,2k-1\right)$ in the Schur basis.

Quiver Hall–Littlewood functions and Kostka–Shoji polynomials

For any triple $(i,a,\mu)$ consisting of a vertex $i$ in a quiver $Q$, a positive integer $a$, and a dominant $GL_a$-weight $\mu$, we define a quiver current $H^{(i,a)}_\mu$ acting on the tensor power $\Lambda^Q$ of symmetric functions over the vertices of $Q$. These provide a quiver generalization of parabolic Garsia-Jing creation operators in the theory of Hall-Littlewood symmetric functions. For a triple $(\mathbf{i},\mathbf{a},\mu(\bullet))$ of sequences of such data, we define the quiver Hall-Littlewood function $H^{\mathbf{i},\mathbf{a}}_{\mu(\bullet)}$ as the result of acting on $1\in\Lambda^Q$ by the corresponding sequence of quiver currents. The quiver Kostka-Shoji polynomials are the expansion coefficients of $H^{\mathbf{i},\mathbf{a}}_{\mu(\bullet)}$ in the tensor Schur basis. These polynomials include the Kostka-Foulkes polynomials and parabolic Kostka polynomials (Jordan quiver) and the Kostka-Shoji polynomials (cyclic quiver) as special cases. We show that the quiver Kostka-Shoji polynomials are graded multiplicities in the equivariant Euler characteristic of a vector bundle on Lusztig's convolution diagram determined by the sequences $\mathbf{i},\mathbf{a}$. For certain compositions of currents we conjecture higher cohomology vanishing of the associated vector bundle on Lusztig's convolution diagram. For quivers with no branching we propose an explicit positive formula for the quiver Kostka-Shoji polynomials in terms of catabolizable multitableaux. We also relate our constructions to $K$-theoretic Hall algebras, by realizing the quiver Kostka-Shoji polynomials as natural structure constants and showing that the quiver currents provide a symmetric function lifting of the corresponding shuffle product. In the case of a cyclic quiver, we explain how the quiver currents arise in Saito's vertex representation of the quantum toroidal algebra of type $\mathfrak{sl}_r$.

Skew key polynomials and a generalized Littlewood–Richardson rule

Young’s lattice is a partial order on integer partitions whose saturated chains correspond to standard Young tableaux, one type of combinatorial object that generates the Schur basis for symmetric functions. Generalizing Young’s lattice, we introduce a new partial order on weak compositions that we call the key poset. Saturated chains in this poset correspond to standard key tableaux, the combinatorial objects that generate the key polynomials, a nonsymmetric polynomial generalization of the Schur basis. Generalizing skew Schur functions, we define skew key polynomials in terms of this new poset. Using weak dual equivalence, we give a nonnegative weak composition Littlewood–Richardson rule for the key expansion of skew key polynomials, generalizing the flagged Littlewood–Richardson rule of Reiner and Shimozono.

On the vertex operator representation of Lie algebras of matrices

The polynomial ring Br:=Q[e1,…,er] in r indeterminates is a representation of the Lie algebra of all the endomorphism of Q[X] vanishing at Xj for all but finitely many j. We determine the structural formal power series of the Br-representation of gl∞(Q). This is a formal power series in r+2 indeterminates encoding the images of all the basis elements of Br under the action of the generating function of elementary endomorphisms of Q[X]. The obtained expression implies (and improves) a formula by Gatto & Salehyan, which only computes the generating functions for the images of specified basis elements. For sake of completeness, in the last section we construct the structural formal power series of the B=B∞-representation of gl∞(Q). It consists in the evaluation of a bosonic vertex operator against the generating function of the standard Schur basis of B. It provides an alternative description of the bosonic representation of gl∞, due to Date, Jimbo, Kashiwara and Miwa, which does not explicitly involve exponentials of differential operators.

The Prism tableau model for Schubert polynomials

The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1; x2; : : :]. We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Gr¨obner geometry of matrix Schubert varieties.

Pentad and triangular structures behind the Racah matrices

Somewhat unexpectedly, the study of the family of twisted knots revealed a hidden structure behind exclusive Racah matrices $$\bar{S}$$, which control non-associativity of the representation product in a peculiar channel $$R\otimes \bar{R} \otimes R \longrightarrow R$$. These $$\bar{S}$$ are simultaneously symmetric and orthogonal and therefore admit two decompositions: as quadratic forms, $$\bar{S} \sim \mathcal{E}^{tr}\mathcal{E}$$, and as operators: $$\bar{T}\bar{S}\bar{T} = S T^{-1} S^{-1}$$. Here, $$\bar{T}$$ and T consist of the eigenvalues of the quantum $$\mathcal{R}$$-matrices in channels $$R\otimes \bar{R}$$ and $$R\otimes R$$, respectively, S is the second exclusive Racah matrix for $$\bar{R}\otimes R\otimes R \longrightarrow R$$ (still orthogonal, but no longer symmetric) and $$\mathcal{E}$$ is a triangular matrix. It can be further used to construct the KNTZ evolution matrix $$\mathcal{B}=\mathcal{E}\bar{T}^2\mathcal{E}^{-1}$$, which is also triangular and explicitly expressible through the skew Schur and Macdonald functions—what makes Racah matrices calculable. Moreover, $$\mathcal{B}$$ is somewhat similar to Ruijsenaars Hamiltonian, which is used to define Macdonald functions, and gets triangular in the Schur basis. Discovery of this pentad structure $$(\bar{T},\bar{S},S,\mathcal{E},\mathcal{B})$$, associated with the universal $$\mathcal{R}$$-matrix, can lead to further insights about representation theory, knot invariants and Macdonald–Kerov functions.

A comment on the combinatorics of the vertex operator $\Gamma _{(t|X)}$

The Jacobi--Trudi identity associates a symmetric function to any integer sequence. Let $\Gamma _{(t|X)}$ be the vertex operator defined by $\Gamma _{(t|X)} s_\alpha =\sum _{n \in \mathbb{Z} } s_{(n,\alpha )} [X] t^n$. We provide a combinatorial proof for the identity $\Gamma _{(t|X)} s_\alpha = \sigma [tX] s_{\alpha }[x-1/t] $ due to Thibon et al. We include an overview of all the combinatorial ideas behind this beautiful identity, including a combinatorial description for the expansion of $s_{(n,\alpha )} [X] $ in the Schur basis, for any integer value of $n$.

Specht modules decompose as alternating sums of restrictions of Schur modules

Schur modules give the irreducible polynomial representations of the general linear group G L t \mathrm {GL}_t . Viewing the symmetric group S t \mathfrak {S}_t as a subgroup of G L t \mathrm {GL}_t , we may restrict Schur modules to S t \mathfrak {S}_t and decompose the result into a direct sum of Specht modules, the irreducible representations of S t \mathfrak {S}_t . We give an equivariant Möbius inversion formula that we use to invert this expansion in the representation ring for S t \mathfrak {S}_t for t t large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternate in sign by degree. In particular, this allows us to define a new basis of symmetric functions whose structure constants are stable Kronecker coefficients and which expand with signs alternating by degree into the Schur basis.

Rigidity for the Hopf Algebra of Quasisymmetric Functions

We investigate the rigidity for the Hopf algebra QSym of quasisymmetric functions with respect to the monomial, the fundamental and the quasisymmetric Schur basis, respectively. By establishing some combinatorial properties of the posets of compositions arising from the analogous Pieri rules for quasisymmetric functions, we show that QSym is rigid as an algebra with respect to the quasisymmetric Schur basis, and rigid as a coalgebra with respect to the monomial and the quasisymmetric Schur basis, respectively. The natural actions of reversal, complement and transpose of the labelling compositions lead to some nontrivial graded (co)algebra automorphisms of QSym. We prove that the linear maps induced by the three actions are precisely the only nontrivial graded algebra automorphisms that take the fundamental basis into itself. Furthermore, the complement map on the labels gives the unique nontrivial graded coalgebra automorphism preserving the fundamental basis, while the reversal map on the labels gives the unique nontrivial graded algebra automorphism preserving the monomial basis. Therefore, QSym is rigid as a Hopf algebra with respect to the monomial and the quasisymmetric Schur basis.

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.

Orthogonal bases of invariants in tensor models

Representation theory provides an efficient framework to count and classify invariants in tensor models of (gauge) symmetry Gd = U(N1) ⊗ · · · ⊗ U(Nd) . We show that there are two natural ways of counting invariants, one for arbitrary Gd and another valid for large rank of Gd. We construct basis of invariant operators based on the counting, and compute correlators of their elements. The basis associated with finite rank of Gd diagonalizes two-point function. It is analogous to the restricted Schur basis used in matrix models. We comment on future directions for investigation.

The Prism tableau model for Schubert polynomials

The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1,x2,…]. We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Gröbner geometry of matrix Schubert varieties.

Sparse multivariate polynomial interpolation on the basis of Schubert polynomials

Schubert polynomials were discovered by A. Lascoux and M. Sch\"utzenberger in the study of cohomology rings of flag manifolds in 1980's. These polynomials generalize Schur polynomials, and form a linear basis of multivariate polynomials. In 2003, Lenart and Sottile introduced skew Schubert polynomials, which generalize skew Schur polynomials, and expand in the Schubert basis with the generalized Littlewood-Richardson coefficients. In this paper we initiate the study of these two families of polynomials from the perspective of computational complexity theory. We first observe that skew Schubert polynomials, and therefore Schubert polynomials, are in $\CountP$ (when evaluating on non-negative integral inputs) and $\VNP$. Our main result is a deterministic algorithm that computes the expansion of a polynomial $f$ of degree $d$ in $\Z[x_1, \dots, x_n]$ in the basis of Schubert polynomials, assuming an oracle computing Schubert polynomials. This algorithm runs in time polynomial in $n$, $d$, and the bit size of the expansion. This generalizes, and derandomizes, the sparse interpolation algorithm of symmetric polynomials in the Schur basis by Barvinok and Fomin (Advances in Applied Mathematics, 18(3):271--285). In fact, our interpolation algorithm is general enough to accommodate any linear basis satisfying certain natural properties. Applications of the above results include a new algorithm that computes the generalized Littlewood-Richardson coefficients.

Rational Parking Functions and Catalan Numbers

The “classical” parking functions, counted by the Cayley number (n+1) n−1, carry a natural permutation representation of the symmetric group S n in which the number of orbits is the Catalan number $${\frac{1}{n+1} \left( \begin{array}{ll} 2n \\ n \end{array} \right)}$$ . In this paper, we will generalize this setup to “rational” parking functions indexed by a pair (a, b) of coprime positive integers. These parking functions, which are counted by b a−1, carry a permutation representation of S a in which the number of orbits is the “rational” Catalan number $${\frac{1}{a+b} \left( \begin{array}{ll} a+b \\ a \end{array} \right)}$$ . First, we compute the Frobenius characteristic of the S a -module of (a, b)-parking functions, giving explicit expansions of this symmetric function in the complete homogeneous basis, the power-sum basis, and the Schur basis. Second, we study q-analogues of the rational Catalan numbers, conjecturing new combinatorial formulas for the rational q-Catalan numbers $${\frac{1}{[a+b]_{q}} {{\left[ \begin{array}{ll} a+b \\ a \end{array} \right]}_{q}}}$$ and for the q-binomial coefficients $${{{\left[ \begin{array}{ll} n \\ k \end{array} \right]}_{q}}}$$ . We give a bijective explanation of the division by [a+b] q that proves the equivalence of these two conjectures. Third, we present combinatorial definitions for q, t-analogues of rational Catalan numbers and parking functions, generalizing the Shuffle Conjecture for the classical case. We present several conjectures regarding the joint symmetry and t = 1/q specializations of these polynomials. An appendix computes these polynomials explicitly for small values of a and b.

A Schur-Like Basis of NSym Defined by a Pieri Rule

Recent research on the algebra of non-commutative symmetric functions and the dual algebra of quasi-symmetric functions has explored some natural analogues of the Schur basis of the algebra of symmetric functions. We introduce a new basis of the algebra of non-commutative symmetric functions using a right Pieri rule. The commutative image of an element of this basis indexed by a partition equals the element of the Schur basis indexed by the same partition and the commutative image is $0$ otherwise. We establish a rule for right-multiplying an arbitrary element of this basis by an arbitrary element of the ribbon basis, and a Murnaghan-Nakayama-like rule for this new basis. Elements of this new basis indexed by compositions of the form $(1^n, m, 1^r)$ are evaluated in terms of the complete homogeneous basis and the elementary basis.

Combinatorial Expansions in $K$-Theoretic Bases

We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape. Included in this class are the Hall-Littlewood polynomials, $k$-Schur functions, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of $GL_n$, Grothendieck functions $\{G_\lambda\}$ represent the $K$-theory of the same space. In this paper, we give a combinatorial description of the coefficients when any element of $\mathcal C$ is expanded in the $G$-basis or the basis dual to $\{G_\lambda\}$.

EQUIVARIANT CHERN CLASSES AND LOCALIZATION THEOREM

For a complex variety with a torus action we propose a new method of computing Chern-Schwartz-MacPherson classes. The method does not apply resolution of singularities. It is based on the Localization Theorem in equivariant cohomology. Equivariant cohomology is a powerful tool for studying complex mani- folds equipped with a torus action. The Localization Theorem of Atiyah and Bott and the resulting formula of Berline-Vergne allow to compute global invariants, for example invariants of singular subsets, in terms of some data attached to the fixed points of the action. We will concentrate on the equi- variant Chern-Schwartz-MacPherson classes. The global class is determined by the local contributions coming from the fixed points. On the other hand, the sum of the local contributions divided by the Euler classes is equal to zero in an appropriate localization of equivariant cohomology. Especially for Grassmannians we obtain interesting formulas with nontrivial relations involving rational functions. We discuss the issue of positivity: the local equivariant Chern class may be presented in various ways, depending on the choice of generating circles of the torus. For some choices we find that the coecients of the presentation are nonnegative. Also the coecients in the Schur basis are nonnegative in many examples, but it turns out that not always.