We introduce a general method for constructing modules for $0$-Hecke algebras and supermodules for $0$-Hecke-Clifford algebras from diagrams of boxes in the plane, and give formulas for the images of …
We introduce a general method for constructing modules for $0$-Hecke algebras and supermodules for $0$-Hecke-Clifford algebras from diagrams of boxes in the plane, and give formulas for the images of these modules in the algebras of quasisymmetric functions and peak functions under the relevant characteristic map. As initial applications, we resolve a question of Jing and Li (2015), introduce a new basis of the peak algebra analogous to the quasisymmetric Schur functions, uncover a new connection between Schur $Q$-functions and quasisymmetric Schur functions, give a representation-theoretic interpretation of families of tableaux used in constructing certain functions in the peak algebra, and establish a common framework for known $0$-Hecke module interpretations of bases of quasisymmetric functions.
Abstract The structure of a 0-Hecke algebra H of type ( W, R ) over a field is examined. H has 2 n distinct irreducible representations, where n = ∣ …
Abstract The structure of a 0-Hecke algebra H of type ( W, R ) over a field is examined. H has 2 n distinct irreducible representations, where n = ∣ R ∣, all of which are one-dimensional, and correspond in a natural way with subsets of R. H can be written as a direct sum of 2 n indecomposable left ideals, in a similar way to Solomon's (1968) decomposition of the underlying Coxeter group W .
Let $(W, I)$ be a finite Coxeter group. In the case where $W$ is a Weyl group, Berenstein and Kazhdan in \cite{BK} constructed a monoid structure on the set of …
Let $(W, I)$ be a finite Coxeter group. In the case where $W$ is a Weyl group, Berenstein and Kazhdan in \cite{BK} constructed a monoid structure on the set of all subsets of $I$ using unipotent $χ$-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type.
Here the Kahler form a, cf. (2.24), is viewed as a (1, I)-form. Moreover, the integrand in 0.3) is equal to a universal constant times the Laplacian applied to (3.l), …
Here the Kahler form a, cf. (2.24), is viewed as a (1, I)-form. Moreover, the integrand in 0.3) is equal to a universal constant times the Laplacian applied to (3.l), plus terms involving at most second order derivatives of the metric. This follows from the formulas for E2 and E4 of Gilkey [30, p. 610], using that, in the computation of the supertrace, the linear contributions of the scalar curvature drop out. So the integrand in (1.3) involves fourth order derivatives of a.
In general, if the manifold is not Kähler, then the Dolbeault-Dirac operator D = 2 $$(\bar{\partial} + \bar{\partial}^{*})$$ is not the most suitable one for getting explicit formulas for (2.39) …
In general, if the manifold is not Kähler, then the Dolbeault-Dirac operator D = 2 $$(\bar{\partial} + \bar{\partial}^{*})$$ is not the most suitable one for getting explicit formulas for (2.39) and (2.40). For instance, if M is a complex analytic manifold and n = 2, then Gilkey [29, Thm. 3.7] proved that the difference $$\rm{trace}_{\bf{c}}\,\it{K}^{+}_{1} (\it{x}) - \rm{trace}_{\bf{c}}\,\it{K}^{-}_{1} (\it{x})$$ of the traces of the coefficients of t−1 in the asymptotic expansion (1.2) is equal to a universal constant times 3.1 $$\rm{d}\,\bar{\partial} \sigma\,=\,\partial \bar{\partial} \sigma\,=\,\partial\,\rm{d}\,\sigma.$$
We begin by deriving an action of the 0-Hecke algebra on standard reverse composition tableaux and use it to discover 0-Hecke modules whose quasisymmetric characteristics are the natural refinements of …
We begin by deriving an action of the 0-Hecke algebra on standard reverse composition tableaux and use it to discover 0-Hecke modules whose quasisymmetric characteristics are the natural refinements of Schur functions known as quasisymmetric Schur functions. Furthermore, we classify combinatorially which of these 0-Hecke modules are indecomposable. From here, we establish that the natural equivalence relation arising from our 0-Hecke action has equivalence classes that are isomorphic to subintervals of the weak Bruhat order on the symmetric group. Focussing on the equivalence classes containing a canonical tableau we discover a new basis for the Hopf algebra of quasisymmetric functions, and use the cardinality of these equivalence classes to establish new enumerative results on truncated shifted reverse tableau studied by Panova and Adin-King-Roichman. Generalizing our 0-Hecke action to one on skew standard reverse composition tableaux, we derive 0-Hecke modules whose quasisymmetric characteristics are the skew quasisymmetric Schur functions of Bessenrodt et al. This enables us to prove a restriction rule that reflects the coproduct formula for quasisymmetric Schur functions, which in turn yields a quasisymmetric branching rule analogous to the classical branching rule for Schur functions.
We begin by deriving an action of the 0-Hecke algebra on standard reverse composition tableaux and use it to discover 0-Hecke modules whose quasisymmetric characteristics are the natural refinements of …
We begin by deriving an action of the 0-Hecke algebra on standard reverse composition tableaux and use it to discover 0-Hecke modules whose quasisymmetric characteristics are the natural refinements of Schur functions known as quasisymmetric Schur functions. Furthermore, we classify combinatorially which of these 0-Hecke modules are indecomposable.
From here, we establish that the natural equivalence relation arising from our 0-Hecke action has equivalence classes that are isomorphic to subintervals of the weak Bruhat order on the symmetric group. Focussing on the equivalence classes containing a canonical tableau we discover a new basis for the Hopf algebra of quasisymmetric functions, and use the cardinality of these equivalence classes to establish new enumerative results on truncated shifted reverse tableau studied by Panova and Adin-King-Roichman.
Generalizing our 0-Hecke action to one on skew standard reverse composition tableaux, we derive 0-Hecke modules whose quasisymmetric characteristics are the skew quasisymmetric Schur functions of Bessenrodt et al. This enables us to prove a restriction rule that reflects the coproduct formula for quasisymmetric Schur functions, which in turn yields a quasisymmetric branching rule analogous to the classical branching rule for Schur functions.
The Clifford action on superspaces is analyzed with a view on generalized Dirac fields taking values in some Clifford supermodule. the stress is here on two principles: complexification and polarisation. …
The Clifford action on superspaces is analyzed with a view on generalized Dirac fields taking values in some Clifford supermodule. the stress is here on two principles: complexification and polarisation. For applications in field theory, the underlying vector space may carry either a Euclidean or a Minkowskian structure.
We study $\phi_\epsilon$-coordinated modules for vertex algebras, where $\phi_\epsilon$ with $\epsilon$ an integer parameter is a family of associates of the one-dimensional additive formal group. As the main results, we …
We study $\phi_\epsilon$-coordinated modules for vertex algebras, where $\phi_\epsilon$ with $\epsilon$ an integer parameter is a family of associates of the one-dimensional additive formal group. As the main results, we obtain a Jacobi type identity and a commutator formula for $\phi_\epsilon$-coordinated modules. We then use these results to study $\phi_\epsilon$-coordinated modules for vertex algebras associated to Novikov algebras by Primc.
In this paper, we study the relation between the cocenter $\overline{\tilde \ch_0}$ and the finite dimensional representations of an affine $0$-Hecke algebra $\tilde \ch_0$. As a consequence, we obtain a …
In this paper, we study the relation between the cocenter $\overline{\tilde \ch_0}$ and the finite dimensional representations of an affine $0$-Hecke algebra $\tilde \ch_0$. As a consequence, we obtain a new criterion on the supersingular modules: a (virtual) module of $\tilde \ch_0$ is supersingular if and only if its character vanishes on the non-supersingular part of $\overline{\tilde \ch_0}$.
In the present paper we determine the representation type of the 0-Hecke algebra of a finite Coxeter group.
In the present paper we determine the representation type of the 0-Hecke algebra of a finite Coxeter group.
The task of a theory of Schubert polynomials is to produce explicit representatives for Schubert classes in the cohomology ring of a flag variety, and to do so in a …
The task of a theory of Schubert polynomials is to produce explicit representatives for Schubert classes in the cohomology ring of a flag variety, and to do so in a manner that is as natural as possible from a combinatorial point of view. To explain more fully, let us review a special case, the Schubert calculus for Grassmannians, where one asks for the number of linear spaces of given dimension satisfying certain geometric conditions. A typical problem is to find the number of lines meeting four given lines in general position in 3-space (answer below). For each of the four given lines, the set of lines meeting it is a Schubert variety in the Grassmannian and we want the number of intersection points of these four subvarieties. In the modem solution of this problem, the Schubert varieties induce canonical elements of the cohomology ring of the Grassmannian, called Schubert classes. The product of these Schubert classes is the class of a point times the number of intersection points, counted with appropriate multiplicities. This reformulation of the problem, though one of the great achievements of algebraic geometry, is only part of a solution. It remains to give a concrete model for the cohomology ring that makes explicit computation with Schubert classes possible. As it happens, the cohomology rings of Grassmannians can be identified with quotients of a polynomial ring so that Schubert classes correspond to Schur functions. Intersection numbers such as we are considering then turn out to be Littlewood-Richardson coefficients. For example, the answer to our four-lines 4 problem is the coefficient of the Schur function s(2 2) in the product s(1) X or 2. For an extended treatment and history of the subject, see [10], [11], [18]. The identification of Schur functions as Schubert polynomials for Grassmannians is a consequence of a more general and now highly developed theory of Schubert polynomials for the flag varieties of the special linear groups SL(n, C). The starting point for this more general theory is a construction of
This article is devoted to the study of several algebras related to symmetric functions, which admit linear bases labelled by various combinatorial objects: permutations (free quasi-symmetric functions), standard Young tableaux …
This article is devoted to the study of several algebras related to symmetric functions, which admit linear bases labelled by various combinatorial objects: permutations (free quasi-symmetric functions), standard Young tableaux (free symmetric functions) and packed integer matrices (matrix quasi-symmetric functions). Free quasi-symmetric functions provide a kind of noncommutative Frobenius characteristic for a certain category of modules over the 0-Hecke algebras. New examples of indecomposable H n (0)-modules are discussed, and the homological properties of H n (0) are computed for small n. Finally, the algebra of matrix quasi-symmetric functions is interpreted as a convolution algebra.
Abstract The structure of a 0-Hecke algebra H of type ( W, R ) over a field is examined. H has 2 n distinct irreducible representations, where n = ∣ …
Abstract The structure of a 0-Hecke algebra H of type ( W, R ) over a field is examined. H has 2 n distinct irreducible representations, where n = ∣ R ∣, all of which are one-dimensional, and correspond in a natural way with subsets of R. H can be written as a direct sum of 2 n indecomposable left ideals, in a similar way to Solomon's (1968) decomposition of the underlying Coxeter group W .
Let be the tensor algebra of the identity representation of the Lie superalgebras in the series and . The method of Weyl is used to construct a correspondence between the …
Let be the tensor algebra of the identity representation of the Lie superalgebras in the series and . The method of Weyl is used to construct a correspondence between the irreducible representations (respectively, the irreducible projective representations) of the symmetric group and the irreducible -(respectively, -) submodules of . The properties of the representations are studied on the basis of this correspondence. A formula is given for the characters on the irreducible -submodules of . Bibliography: 8 titles.
An (ordinary)<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"><mml:semantics><mml:mi>P</mml:mi><mml:annotation encoding="application/x-tex">P</mml:annotation></mml:semantics></mml:math></inline-formula>-partition is an order-preserving map from a partially ordered set to a chain, with special rules specifying where equal values may occur. Examples include …
An (ordinary)<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"><mml:semantics><mml:mi>P</mml:mi><mml:annotation encoding="application/x-tex">P</mml:annotation></mml:semantics></mml:math></inline-formula>-partition is an order-preserving map from a partially ordered set to a chain, with special rules specifying where equal values may occur. Examples include number-theoretic partitions (ordered and unordered, strict or unrestricted), plane partitions, and the semistandard tableaux associated with Schur’s<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"><mml:semantics><mml:mi>S</mml:mi><mml:annotation encoding="application/x-tex">S</mml:annotation></mml:semantics></mml:math></inline-formula>-functions. In this paper, we introduce and develop a theory of enriched<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"><mml:semantics><mml:mi>P</mml:mi><mml:annotation encoding="application/x-tex">P</mml:annotation></mml:semantics></mml:math></inline-formula>-partitions; like ordinary<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"><mml:semantics><mml:mi>P</mml:mi><mml:annotation encoding="application/x-tex">P</mml:annotation></mml:semantics></mml:math></inline-formula>-partitions, these are order-preserving maps from posets to chains, but with different rules governing the occurrence of equal values. The principal examples of enriched<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"><mml:semantics><mml:mi>P</mml:mi><mml:annotation encoding="application/x-tex">P</mml:annotation></mml:semantics></mml:math></inline-formula>-partitions given here are the tableaux associated with Schur’s<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q"><mml:semantics><mml:mi>Q</mml:mi><mml:annotation encoding="application/x-tex">Q</mml:annotation></mml:semantics></mml:math></inline-formula>-functions. In a sequel to this paper, further applications related to commutation monoids and reduced words in Coxeter groups will be presented.
We associate a polynomial to any diagram of unit cells in the first quadrant of the plane using Kohnert's algorithm for moving cells down. In this way, for every weak …
We associate a polynomial to any diagram of unit cells in the first quadrant of the plane using Kohnert's algorithm for moving cells down. In this way, for every weak composition one can choose a cell diagram with corresponding row-counts, with each choice giving rise to a combinatorially-defined basis of polynomials. These Kohnert bases provide a simultaneous generalization of Schubert polynomials and Demazure characters for the general linear group. Using the monomial and fundamental slide bases defined earlier by the authors, we show that Kohnert polynomials stabilize to quasisymmetric functions that are nonnegative on the fundamental basis for quasisymmetric functions. For initial applications, we define and study two new Kohnert bases. The elements of one basis are conjecturally Schubert-positive and stabilize to the skew-Schur functions; the elements of the other basis stabilize to a new basis of quasisymmetric functions that contains the Schur functions.
Recently Tewari and van Willigenburg constructed modules of the 0-Hecke algebra that are mapped to the quasisymmetric Schur functions by the quasisymmetric characteristic and decomposed them into a direct sum …
Recently Tewari and van Willigenburg constructed modules of the 0-Hecke algebra that are mapped to the quasisymmetric Schur functions by the quasisymmetric characteristic and decomposed them into a direct sum of certain submodules. We show that these submodules are indecomposable by determining their endomorphism rings.
The extended Schur functions form a basis of quasisymmetric functions that contains the Schur functions. We provide a representation-theoretic interpretation of this basis by constructing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> …
The extended Schur functions form a basis of quasisymmetric functions that contains the Schur functions. We provide a representation-theoretic interpretation of this basis by constructing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Hecke modules whose quasisymmetric characteristics are the extended Schur functions. We further prove these modules are indecomposable.
We construct indecomposable modules for the $0$-Hecke algebra whose characteristics are the dual immaculate basis of the quasi-symmetric functions.
We construct indecomposable modules for the $0$-Hecke algebra whose characteristics are the dual immaculate basis of the quasi-symmetric functions.
We introduce a new basis of the non-commutative symmetric functions whose commutative images are Schur functions. Dually, we build a basis of the quasi-symmetric functions which expand positively in the …
We introduce a new basis of the non-commutative symmetric functions whose commutative images are Schur functions. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions. We then use the basis to construct a non-commutative lift of the Hall-Littlewood symmetric functions with similar properties to their commutative counterparts.
Abstract Let n be a nonnegative integer. For each composition $\alpha $ of n , Berg, Bergeron, Saliola, Serrano and Zabrocki introduced a cyclic indecomposable $H_n(0)$ -module $\mathcal {V}_{\alpha }$ …
Abstract Let n be a nonnegative integer. For each composition $\alpha $ of n , Berg, Bergeron, Saliola, Serrano and Zabrocki introduced a cyclic indecomposable $H_n(0)$ -module $\mathcal {V}_{\alpha }$ with a dual immaculate quasisymmetric function as the image of the quasisymmetric characteristic. In this paper, we study $\mathcal {V}_{\alpha }$ s from the homological viewpoint. To be precise, we construct a minimal projective presentation of $\mathcal {V}_{\alpha }$ and a minimal injective presentation of $\mathcal {V}_{\alpha }$ as well. Using them, we compute $\mathrm {Ext}^1_{H_n(0)}(\mathcal {V}_{\alpha }, \mathbf {F}_{\beta })$ and $\mathrm {Ext}^1_{H_n(0)}( \mathbf {F}_{\beta }, \mathcal {V}_{\alpha })$ , where $\mathbf {F}_{\beta }$ is the simple $H_n(0)$ -module attached to a composition $\beta $ of n . We also compute $\mathrm {Ext}_{H_n(0)}^i(\mathcal {V}_{\alpha },\mathcal {V}_{\beta })$ when $i=0,1$ and $\beta \le _l \alpha $ , where $\le _l$ represents the lexicographic order on compositions.
Abstract We extend the notion of ascent-compatibility from symmetric groups to all Coxeter groups, thereby providing a type-independent framework for constructing families of modules of $0$ -Hecke algebras. We apply …
Abstract We extend the notion of ascent-compatibility from symmetric groups to all Coxeter groups, thereby providing a type-independent framework for constructing families of modules of $0$ -Hecke algebras. We apply this framework in type B to give representation–theoretic interpretations of a number of noteworthy families of type- B quasisymmetric functions. Next, we construct modules of the type- B $0$ -Hecke algebra corresponding to type- B analogs of Schur functions and introduce a type- B analog of Schur Q -functions; we prove that these shifted domino functions expand positively in the type- B peak functions. We define a type- B analog of the $0$ -Hecke–Clifford algebra, and we use this to provide representation–theoretic interpretations for both the type- B peak functions and the shifted domino functions. We consider the modules of this algebra induced from type- B $0$ -Hecke modules constructed via ascent-compatibility and prove a general formula, in terms of type- B peak functions, for the type- B quasisymmetric characteristics of the restrictions of these modules.