Jacobi-Trudi Determinants over Finite Fields

It is an important problem in algebraic combinatorics to deduce the Schur function expansion of a symmetric function whose expansion in terms of the fundamental quasisymmetric function is known. For example, formulas are known for the fundamental expansion of a Macdonald symmetric function and for the plethysm of two Schur functions, while the Schur expansions of these expressions are still elusive. Based on work of Egge, Loehr and Warrington, Garsia and Remmel provided a method to obtain the Schur expansion from the fundamental expansion by replacing each quasisymmetric function by a Schur function (not necessarily indexed by a partition) and using straightening rules to obtain the Schur expansion. Here we provide a new method that only involves the coefficients of the quasisymmetric functions indexed by partitions and the quasi-Kostka matrix. As an application, we identify the lexicographically largest term in the Schur expansion of the plethysm of two Schur functions. We provide the Schur expansion of $s_w[s_h](x,y)$ for $w=2,3,4$ using novel symmetric chain decompositions of Young's lattice for partitions in a $w\times h$ box. For $w=4$, this is the first known combinatorial expression for the coefficient of $s_{\lambda}$ in $s_{w}[s_{h}]$ for two-row partitions $\lambda$, and for $w=3$ the combinatorial expression is new.

Powers of staircase Schur functions and symmetric analogues of Bessel polynomials

For the quantum cohomology of Grassmannians, see [1]. Given integers l, N with 1 6 l 0〈σλ, σμ, σν∨〉d q σν , where ν ∨ is the partition dual to ν. Let ζ be a primitive Nth root of (−1). Put K = Q(ζ), k = N − l and let Λ = Z[e1, e2, . . . ] be the (graded) ring of symmetric functions, where ei is the ith elementary symmetric function. Let hi be the ith complete symmetric function. Putting E(t) = P eit , H(t) = P hit , we have E(−t)H(t) = 1. The result of applying a symmetric function σ to a tuple (x1, . . . , xn) of arguments will be denoted by σ(x1, . . . , xn). Finally, we put Λ ′ = Λ/(el+1, el+2, . . . ), ΛQ = Λ ⊗ Q, ΛQ = Λ ′ ⊗ Q and ΛK = ΛQ ⊗Q K. Theorem 1 (Siebert–Tian [2]). The homomorphism of rings ST: Λ[q]→QH(G,Z) defined by ST (q) = 1 ⊗ q, ST (ei) = ci(S) ⊗ 1 is an epimorphism, and KerST = (el+1, el+2, . . . ;hN−l+1, . . . , hN−1, hN + (−1)q). Let QH(G,Q) = QH(G,Z) ⊗ Q be the rational quantum cohomology ring and let QH(G,Q, 1) denote its specialization for q = 1. Putting I1 = (hN−l+1, . . . , hN−1, hN + (−1)),

We present a realization of untwisted vertex operators in terms of operations on Schur functions. Calculations of matrix elements and traces of products of vertex operators are performed using results from the classical theory of symmetric functions. The concepts of compound, composite and supersymmetric Schur functions naturally appear in this context. Furthermore, a trace formula for a product of vertex operators turns out to be a generalization of a MacDonald identity reformulated in terms of Schur functions.

A celebrated result of Farahat and Higman constructs an algebra FH which “interpolates” the centres \(Z(\mathbb {Z}S_{n})\) of group algebras of the symmetric groups Sn. We extend these results from symmetric group algebras to type A Iwahori-Hecke algebras, Hn(q). In particular, we explain how to construct an algebra FHq “interpolating” the centres Z(Hn(q)). We prove that FHq is isomorphic to \(\mathcal {R}[q,q^{-1}] \otimes _{\mathbb {Z}} {\Lambda }\) (where \(\mathcal {R}\) is the ring of integer-valued polynomials, and Λ is the ring of symmetric functions). The isomorphism can be described as “evaluation at Jucys-Murphy elements”, leading to a proof of a conjecture of Francis and Wang. This yields character formulae for the Geck-Rouquier basis of Z(Hn(q)) when acting on Specht modules.

A result of Farahat and Higman shows that there is a “universal” algebra, FH , interpolating the centres of symmetric group algebras, Z ( ℤ S n ) . We explain that this algebra is isomorphic to ℛ ⊗ Λ , where ℛ is the ring of integer-valued polynomials and Λ is the ring of symmetric functions. Moreover, the isomorphism is via “evaluation at Jucys–Murphy elements”, which leads to character formulae for symmetric groups. Then, we generalise this result to wreath products Γ ≀ S n of a fixed finite group Γ . This involves constructing wreath-product versions ℛ Γ and Λ ( Γ * ) of ℛ and Λ , respectively, which are interesting in their own right (for example, both are Hopf algebras). We show that the universal algebra for wreath products, FH Γ , is isomorphic to ℛ Γ ⊗ Λ ( Γ * ) and use this to compute the p -blocks of wreath products.

Skew Weyl Modules for GLn and Degree Reduction for Schur Algebras

A proof of the fermionic theta coinvariant conjecture

Using the vertex operator representations for symplectic and orthogonal Schur functions, we define two families of symmetric functions and show thatthey are the skew symplectic and skew orthogonal Schur polynomials defined implicitly by Koike and Terada and satisfy the general branching rules. Furthermore, we derive the Jacobi-Trudi identities and Gelfand-Tsetlin patterns for these symmetric functions. Additionally, the vertex operator method yields their Cauchy-type identities. This demonstrates that vertex operator representations serve not only as a tool for studying symmetric functions but also offers unified realizations for skew Schur functions of types A, C, and D.

We derive group branching laws for formal characters of subgroups of leaving invariant an arbitrary tensor T? of Young symmetry type ? where ? is an integer partition. The branchings and fixing a vector vi, a symmetric tensor gij = gji and an antisymmetric tensor fij = ?fji, respectively, are obtained as special cases. All new branchings are governed by Schur function series obtained from plethysms of the Schur function s? ? {?} by the basic M series of complete symmetric functions and the L = M?1 series of elementary symmetric functions. Our main technical tool is that of Hopf algebras and our main result is the derivation of a coproduct for any Schur function series obtained by plethysm from another such series. Therefrom one easily obtains ?-generalized Newell?Littlewood formulae and the algebra of the formal group characters of these subgroups is established. Concrete examples and extensive tabulations are displayed for and , showing their involved and nontrivial representation theory. The nature of the subgroups is shown to be in general affine and in some instances non-reductive. We discuss the complexity of the coproduct formula and give a graphical notation to cope with it. We also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly nontrivial 2-cocycles. The algebra of subgroup characters is identified as a cliffordization of the algebra of symmetric functions for formal characters. Modification rules are beyond the scope of the present paper, but are briefly discussed.

Starting with a family of Markov chains corresponding to a trellis (or ladder) model we show how to find symmetric functions in the eigenvalues of the transition matrix. Schur functions arise via minors of powers of the transition matrix. We give a path model for finding the Schur functions in terms of the transition probabilities of the trellis. Limit theorems for powers of the transition matrix yield limit theorems for the exterior powers of the transition matrix which in turn yield limit theorems for corresponding families of symmetric functions. We present a theorem on Ces`aro convergence of powers of a matrix that is the exterior power of a stochastic matrix.

Lam and Pylyavskyy introduced loop symmetric functions as a generalization of symmetric functions. They defined loop Schur functions as generating functions over semistandard tableaux with respect to a `colored weight,' and they proved a Jacobi--Trudi-style determinantal formula for these generating functions. We prove that loop Schur functions can be expressed as a ratio of `loop alternants,' extending the analogy with Schur functions. As an application, we give a new proof of the loop version of the Murnaghan--Nakayama rule.

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 $\{G_\lambda\}$.

Green Functions Associated to Complex Reflection Groups

We establish the conjecture of Reiner and Yong for an explicit combinatorial formula for the expansion of a Grothendieck polynomial into the basis of Lascoux polynomials. This expansion is a subtle refinement of its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, and Yong, which gives the expansion of stable Grothendieck polynomials indexed by permutations into Grassmannian stable Grothendieck polynomials. Our expansion is the K K -theoretic analogue of that of a Schubert polynomial into Demazure characters, whose symmetric analogue is the expansion of a Stanley symmetric function into Schur functions. Our expansions extend to flagged Grothendieck polynomials.