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Journal articles on the topic 'Grassmannian Coordinates'

Author: Grafiati

Published: 7 September 2023

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1

MOHAMMEDI, NOUREDDINE. "FRACTIONAL SUPERSYMMETRY." Modern Physics Letters A 10, no. 18 (June 14, 1995): 1287–91. http://dx.doi.org/10.1142/s021773239500140x.

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A symmetry between bosonic coordinates and some Grassmannian-type coordinates is presented. Commuting two of these Grassmannian-type variables results in an arbitrary phase factor (not just a minus sign). This symmetry is also realized at the level of the field theory.

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2

Dzhunushaliev, V. "A Geometrical Interpretation of Grassmannian Coordinates." General Relativity and Gravitation 34, no. 8 (August 2002): 1267–75. http://dx.doi.org/10.1023/a:1019782619091.

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3

Talaska, Kelli. "Combinatorial formulas for -coordinates in a totally nonnegative Grassmannian." Journal of Combinatorial Theory, Series A 118, no. 1 (January 2011): 58–66. http://dx.doi.org/10.1016/j.jcta.2009.10.006.

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4

DELBOURGO, R., S. TWISK, and R. B. ZHANG. "GRAND UNIFICATION AND GRASSMANNIAN KALUZA-KLEIN THEORY." Modern Physics Letters A 03, no. 11 (September 1988): 1073–78. http://dx.doi.org/10.1142/s0217732388001264.

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It is possible to construct a grand unified model, including gravity, by adjoining further anticommuting coordinates to space-time. We carry out this program for SU(5) and SO(10) unified models; curiously, the former is much more economical than the latter and simply requires five complex Grassmann variables ξm in addition to the four xμ.

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5

Purbhoo, Kevin. "Total Nonnegativity and Stable Polynomials." Canadian Mathematical Bulletin 61, no. 4 (November 20, 2018): 836–47. http://dx.doi.org/10.4153/cmb-2018-006-7.

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AbstractWe consider homogeneous multiaffine polynomials whose coefficients are the Plücker coordinates of a point V of the Grassmannian. We show that such a polynomial is stable (with respect to the upper half plane) if and only if V is in the totally nonnegative part of the Grassmannian. To prove this, we consider an action of matrices on multiaffine polynomials. We show that a matrix A preserves stability of polynomials if and only if A is totally nonnegative. The proofs are applications of classical theory of totally nonnegative matrices, and the generalized Pólya–Schur theory of Borcea and Brändén.

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6

Delbourgo, Robert. "Grassmannian duality and the particle spectrum." International Journal of Modern Physics A 31, no. 26 (September 20, 2016): 1650153. http://dx.doi.org/10.1142/s0217751x16501530.

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Schemes based on anticommuting scalar coordinates, corresponding to properties, lead to generations of particles naturally. The application of Grassmannian duality cuts down the number of states substantially and is vital for constructing sensible Lagrangians anyhow. We apply duality to all of the subgroups within the classification group [Formula: see text], which encompasses the standard model gauge group, and thereby determine the full state inventory; this includes the definite prediction of quarks with charge [Formula: see text] and other exotic states. Assuming universal gravitational coupling to the gauge fields and parity even property curvature, we also obtain [Formula: see text] which is not far from the experimental value around the [Formula: see text] mass.

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7

VISWANATHAN, K. S., and R. PARTHASARATHY. "EXTRINSIC GEOMETRY OF WORLD-SHEET SUPERSYMMETRY THROUGH GENERALIZED SUPER-GAUSS MAPS." International Journal of Modern Physics A 07, no. 24 (September 30, 1992): 5995–6011. http://dx.doi.org/10.1142/s0217751x92002714.

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The extrinsic geometry of N=1 world-sheet supersymmetry is studied through generalized super-Gauss map. The world sheet, realized as a conformally immersed super-Riemann surface S in Rn (n=3 is studied for simplicity) is mapped into the supersymmetric Grassmannian G2,3. In order for the Grassmannian fields to form (super) tangent planes to S, certain integrability conditions are satisfied by G2,n fields. These conditions are explicitly derived. The supersymmetric invariant action for the Kähler σ-model G2,3 is reexpressed in terms of the world-sheet coordinates, thereby an off-shell supersymmetric generalization of the action proportional to the extrinsic curvature of the immersed surface is obtained.

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LOSEV, A., and A. TURBINER. "MULTIDIMENSIONAL EXACTLY SOLVABLE PROBLEMS IN QUANTUM MECHANICS AND PULLBACKS OF AFFINE COORDINATES ON THE GRASSMANNIAN." International Journal of Modern Physics A 07, no. 07 (March 20, 1992): 1449–65. http://dx.doi.org/10.1142/s0217751x92000636.

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Multidimensional exactly solvable problems related to compact hidden-symmetry groups are discussed. Natural coordinates on homogeneous space are introduced. It is shown that a potential and scalar curvature of the problem considered have quite a simple form of quadratic polynomials in these coordinates. A mysterious relation between the potential and the curvature observed for SU(2) in Refs. 2 and 3 is obtained in a simple way.

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9

BENZAIR, H., M. MERAD, and T. BOUDJEDAA. "NONCOMMUTATIVE PATH INTEGRAL FOR SPINLESS RELATIVISTIC EQUATION IN THE TWO-COMPONENT THEORY." Modern Physics Letters A 28, no. 32 (October 6, 2013): 1350144. http://dx.doi.org/10.1142/s0217732313501447.

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In this paper, we have constructed the Green function of the Feshbach–Villars (FV) spinless particle in a noncommutative (NC) phase-space coordinates, where the Pauli matrices describing the charge symmetry are replaced by the Grassmannian odd variables. Subsequently, for the perform calculations, we diagonalize the Hamiltonian governing the dynamics of the system via the Foldy–Wouthuysen (FW) canonical transformation. The exact calculations have been done in the cases of free particle and magnetic field interaction. In both cases, the energy eigenvalues and their corresponding eigenfunctions are deduced.

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10

Fioresi, Rita, María A. Lledó, and Junaid Razzaq. "Quantum Chiral Superfields." Journal of Physics: Conference Series 2531, no. 1 (June 1, 2023): 012015. http://dx.doi.org/10.1088/1742-6596/2531/1/012015.

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Abstract We define the ordinary Minkowski space inside the conformal space according to Penrose and Manin as homogeneous spaces for the Poincaré and conformal group respectively. We realize the supersymmetric (SUSY) generalizations of such homogeneous spaces over the complex and the real fields. We finally investigate chiral (antichiral) superfields, which are superfields on the super Grassmannian, Gr(2|1, 4|1), respectively on Gr(2|0, 4|1). They ultimately give the twistor coordinates necessary to describe the conformal superspace as the flag Fl(2|0, 2|1; 4|1) and the Minkowski superspace as its big cell.

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11

Nieto, Isidro. "Real quartic surfaces containing16skew lines." International Journal of Mathematics and Mathematical Sciences 2004, no. 44 (2004): 2331–45. http://dx.doi.org/10.1155/s0161171204308112.

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It is well known that there is an open three-dimensional subvarietyMsof the Grassmannian of lines inℙ3which parametrizes smooth irreducible complex surfaces of degree 4 which are Heisenberg invariant, and each quartic contains 32 lines but only 16 skew lines, being determined by its configuration of lines, are called adouble 16. We consider here the problem of visualizing in a computer the real Heisenberg invariant quartic surface and the real double 16. We construct a family of pointsl∈Msparametrized by a two-dimensional semialgebraic variety such that under a change of coordinates oflinto its Plüecker, coordinates transform into the real coordinates for a lineLinℙ3, which is then used to construct a program in Maple 7. The program allows us to draw the quartic surface and the set of transversal lines toL. Additionally, we include a table of a group of examples. For each test example we specify a parameter, the viewing angle of the image, compilation time, and other visual properties of the real surface and its real double 16. We include at the end of the paper an example showing the surface containing the double 16.

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12

Knutson, Allen, Thomas Lam, and David E. Speyer. "Positroid varieties: juggling and geometry." Compositio Mathematica 149, no. 10 (August 19, 2013): 1710–52. http://dx.doi.org/10.1112/s0010437x13007240.

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AbstractWhile the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown–Goodearl–Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from [A. Knutson, T. Lam and D. Speyer, Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG]], we show that positroid varieties are normal, Cohen–Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov’s and Buch–Kresch–Tamvakis’ approaches to quantum Schubert calculus.

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Srinivas, N., and R. P. Malik. "Nilpotent symmetries and Curci–Ferrari-type restrictions in 2D non-Abelian gauge theory: Superfield approach." International Journal of Modern Physics A 32, no. 33 (November 30, 2017): 1750193. http://dx.doi.org/10.1142/s0217751x17501937.

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We derive the off-shell nilpotent symmetries of the two [Formula: see text]-dimensional (2D) non-Abelian 1-form gauge theory by using the theoretical techniques of the geometrical superfield approach to Becchi–Rouet–Stora–Tyutin (BRST) formalism. For this purpose, we exploit the augmented version of superfield approach (AVSA) and derive theoretically useful nilpotent (anti-)BRST, (anti-)co-BRST symmetries and Curci–Ferrari (CF)-type restrictions for the self-interacting 2D non-Abelian 1-form gauge theory (where there is no interaction with matter fields). The derivation of the (anti-)co-BRST symmetries and all possible CF-type restrictions are completely novel results within the framework of AVSA to BRST formalism where the ordinary 2D non-Abelian theory is generalized onto an appropriately chosen [Formula: see text]-dimensional supermanifold. The latter is parametrized by the superspace coordinates [Formula: see text] where [Formula: see text] (with [Formula: see text]) are the bosonic coordinates and a pair of Grassmannian variables [Formula: see text] obey the relationships: [Formula: see text], [Formula: see text]. The topological nature of our 2D theory allows the existence of a tower of CF-type restrictions.

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14

SCOTT, JOSHUA S. "GRASSMANNIANS AND CLUSTER ALGEBRAS." Proceedings of the London Mathematical Society 92, no. 2 (February 20, 2006): 345–80. http://dx.doi.org/10.1112/s0024611505015571.

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This paper follows the program of study initiated by S. Fomin and A. Zelevinsky, and demonstrates that the homogeneous coordinate ring of the Grassmannian $\mathbb{G}(k, n)$ is a {\it cluster algebra of geometric type}. Those Grassmannians that are of {\it finite cluster type} are identified and their cluster variables are interpreted geometrically in terms of configurations of points in $\mathbb{C}\mathbb{P}^2$.

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15

MALIK, R. P. "NILPOTENT SYMMETRY INVARIANCE IN THE SUPERFIELD FORMULATION: THE (NON-)ABELIAN 1-FORM GAUGE THEORIES." International Journal of Modern Physics A 23, no. 22 (September 10, 2008): 3685–705. http://dx.doi.org/10.1142/s0217751x08041591.

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We capture the off-shell as well as the on-shell nilpotent Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetry invariance of the Lagrangian densities of the four (3+1)-dimensional (4D) (non-)Abelian 1-form gauge theories within the framework of the superfield formalism. In particular, we provide the geometrical interpretations for (i) the above nilpotent symmetry invariance, and (ii) the above Lagrangian densities, in the language of the specific quantities defined in the domain of the above superfield formalism. Some of the subtle points, connected with the 4D (non-)Abelian 1-form gauge theories, are clarified within the framework of the above superfield formalism where the 4D ordinary gauge theories are considered on the (4, 2)-dimensional supermanifold parametrized by the four space–time coordinates xμ (with μ = 0, 1, 2, 3) and a pair of Grassmannian variables θ and [Formula: see text]. One of the key results of our present investigation is a great deal of simplification in the geometrical understanding of the nilpotent (anti-)BRST symmetry invariance.

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16

Srinivas, N., T. Bhanja, and R. P. Malik. "(Anti)chiral Superfield Approach to Nilpotent Symmetries: Self-Dual Chiral Bosonic Theory." Advances in High Energy Physics 2017 (2017): 1–14. http://dx.doi.org/10.1155/2017/6138263.

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We exploit the beauty and strength of the symmetry invariant restrictions on the (anti)chiral superfields to derive the Becchi-Rouet-Stora-Tyutin (BRST), anti-BRST, and (anti-)co-BRST symmetry transformations in the case of a two (1+1)-dimensional (2D) self-dual chiral bosonic field theory within the framework of augmented (anti)chiral superfield formalism. Our 2D ordinary theory is generalized onto a (2,2)-dimensional supermanifold which is parameterized by the superspace variable ZM=xμ,θ,θ¯, where xμ (with μ=0,1) are the ordinary 2D bosonic coordinates and (θ,θ¯) are a pair of Grassmannian variables with their standard relationships: θ2=θ¯2=0, θθ¯+θ¯θ=0. We impose the (anti-)BRST and (anti-)co-BRST invariant restrictions on the (anti)chiral superfields (defined on the (anti)chiral (2,1)-dimensional supersubmanifolds of the above general (2,2)-dimensional supermanifold) to derive the above nilpotent symmetries. We do not exploit the mathematical strength of the (dual-)horizontality conditions anywhere in our present investigation. We also discuss the properties of nilpotency, absolute anticommutativity, and (anti-)BRST and (anti-)co-BRST symmetry invariance of the Lagrangian density within the framework of our augmented (anti)chiral superfield formalism. Our observation of the absolute anticommutativity property is a completely novel result in view of the fact that we have considered only the (anti)chiral superfields in our present endeavor.

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17

Bhanja, T., N. Srinivas, and R. P. Malik. "Nilpotent charges of a toy model of Hodge theory and an 𝒩 = 2 SUSY quantum mechanical model: (Anti-)chiral supervariable approach." International Journal of Modern Physics A 34, no. 30 (October 30, 2019): 1950183. http://dx.doi.org/10.1142/s0217751x19501835.

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We derive the nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations for the system of a toy model of Hodge theory (i.e. a rigid rotor) by exploiting the (anti-)BRST and (anti-)co-BRST invariant restrictions on the (anti-)chiral supervariables that are defined on the appropriately chosen [Formula: see text]-dimensional super-submanifolds of the general [Formula: see text]-dimensional supermanifold on which our system of a one [Formula: see text]-dimensional (1D) toy model of Hodge theory is considered within the framework of the augmented version of the (anti-)chiral supervariable approach (ACSA) to Becchi–Rouet–Stora–Tyutin (BRST) formalism. The general [Formula: see text]-dimensional supermanifold is parametrized by the superspace coordinates [Formula: see text], where [Formula: see text] is the bosonic evolution parameter and [Formula: see text] are the Grassmannian variables which obey the standard fermionic relationships: [Formula: see text], [Formula: see text]. We provide the geometrical interpretations for the symmetry invariance and nilpotency property. Furthermore, in our present endeavor, we establish the property of absolute anticommutativity of the conserved fermionic charges which is a completely novel and surprising observation in our present endeavor where we have considered only the (anti-)chiral supervariables. To corroborate the novelty of the above observation, we apply this ACSA to an [Formula: see text] SUSY quantum mechanical (QM) system of a free particle and show that the [Formula: see text] SUSY conserved and nilpotent charges do not absolutely anticommute.

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18

Rao, A. K., A. Tripathi, and R. P. Malik. "Supervariable and BRST Approaches to a Reparameterization Invariant Nonrelativistic System." Advances in High Energy Physics 2021 (July 22, 2021): 1–20. http://dx.doi.org/10.1155/2021/5593434.

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We exploit the theoretical strength of the supervariable and Becchi-Rouet-Stora-Tyutin (BRST) formalisms to derive the proper (i.e., off-shell nilpotent and absolutely anticommuting) (anti-)BRST symmetry transformations for the reparameterization invariant model of a nonrelativistic (NR) free particle whose space x and time t variables are a function of an evolution parameter τ . The infinitesimal reparameterization (i.e., 1D diffeomorphism) symmetry transformation of our theory is defined w.r.t. this evolution parameter τ . We apply the modified Bonora-Tonin (BT) supervariable approach (MBTSA) as well as the (anti)chiral supervariable approach (ACSA) to BRST formalism to discuss various aspects of our present system. For this purpose, our 1D ordinary theory (parameterized by τ ) is generalized onto a 1 , 2 -dimensional supermanifold which is characterized by the superspace coordinates Z M = τ , θ , θ ¯ where a pair of the Grassmannian variables satisfy the fermionic relationships: θ 2 = θ ¯ 2 = 0 , θ θ ¯ + θ ¯ θ = 0 , and τ is the bosonic evolution parameter. In the context of ACSA, we take into account only the 1 , 1 -dimensional (anti)chiral super submanifolds of the general 1 , 2 -dimensional supermanifold. The derivation of the universal Curci-Ferrari- (CF-) type restriction, from various underlying theoretical methods, is a novel observation in our present endeavor. Furthermore, we note that the form of the gauge-fixing and Faddeev-Popov ghost terms for our NR and non-SUSY system is exactly the same as that of the reparameterization invariant SUSY (i.e., spinning) and non-SUSY (i.e., scalar) relativistic particles. This is a novel observation, too.

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19

BAUR, KARIN, DUSKO BOGDANIC, and ANA GARCIA ELSENER. "CLUSTER CATEGORIES FROM GRASSMANNIANS AND ROOT COMBINATORICS." Nagoya Mathematical Journal 240 (June 3, 2019): 322–54. http://dx.doi.org/10.1017/nmj.2019.14.

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The category of Cohen–Macaulay modules of an algebra $B_{k,n}$ is used in Jensen et al. (A categorification of Grassmannian cluster algebras, Proc. Lond. Math. Soc. (3) 113(2) (2016), 185–212) to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$-planes in $n$-space. In this paper, we find canonical Auslander–Reiten sequences and study the Auslander–Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen–Macaulay modules of arbitrary rank. We then use our results to establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the associated Kac–Moody algebra in the tame cases.

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20

Poëls, Anthony. "A class of maximally singular sets for rational approximation." International Journal of Number Theory 16, no. 09 (June 16, 2020): 2005–12. http://dx.doi.org/10.1142/s1793042120501031.

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We say that a subset of [Formula: see text] is maximally singular if its contains points with [Formula: see text]-linearly independent homogenous coordinates whose uniform exponent of simultaneous rational approximation is equal to [Formula: see text], the maximal possible value. In this paper, we give a criterion which provides many such sets including Grassmannians. We also recover a result of the author and Roy about a class of quadratic hypersurfaces.

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21

Akin, Kaan, and David A. Buchsbaum. "A note on the Poincaré resolution of the coordinate ring of the Grassmannian." Journal of Algebra 152, no. 2 (November 1992): 427–33. http://dx.doi.org/10.1016/0021-8693(92)90040-s.

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22

Sasaki, Takeshi, and Masaaki Yoshida. "Invariant theory for linear differential systems modeled after the grassmannian Gr(n, 2n)." Nagoya Mathematical Journal 171 (2003): 163–86. http://dx.doi.org/10.1017/s0027763000025551.

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AbstractWe find invariants for the differential systems of rank 2n in n2 variables with n unknowns under the linear changes of the unknowns with variable coefficients. We look for a set of coefficients that determines the other coefficients, and give transformation rules under the linear changes above and coordinate changes. These can be considered as a generalization of the Schwarzian derivative, which is the invariant for second order ordinary differential equations under the change of the unknown by multiplying a non-zero function. Special treatment is done when n = 2: the conformal structure obtained through the Plücker embedding is studied, and a relation with line congruences is discussed.

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23

Cotti, Giordano, and Davide Guzzetti. "Analytic geometry of semisimple coalescent Frobenius structures." Random Matrices: Theory and Applications 06, no. 04 (October 2017): 1740004. http://dx.doi.org/10.1142/s2010326317400044.

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We present some results of a joint paper with Dubrovin (see references), as exposed at the Workshop “Asymptotic and Computational Aspects of Complex Differential Equations” at the CRM in Pisa, in February 2017. The analytical description of semisimple Frobenius manifolds is extended at semisimple coalescence points, namely points with some coalescing canonical coordinates although the corresponding Frobenius algebra is semisimple. After summarizing and revisiting the theory of the monodromy local invariants of semisimple Frobenius manifolds, as introduced by Dubrovin, it is shown how the definition of monodromy data can be extended also at semisimple coalescence points. Furthermore, a local Isomonodromy theorem at semisimple coalescence points is presented. Some examples of computation are taken from the quantum cohomologies of complex Grassmannians.

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24

Bhanja, T., D. Shukla, and R. P. Malik. "Superspace Unitary Operator in Superfield Approach to Non-Abelian Gauge Theory with Dirac Fields." Advances in High Energy Physics 2016 (2016): 1–11. http://dx.doi.org/10.1155/2016/6367545.

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Within the framework of augmented version of the superfield approach to Becchi-Rouet-Stora-Tyutin (BRST) formalism, we derive the superspace unitary operator (and its Hermitian conjugate) in the context of four (3 + 1)-dimensional (4D) interacting non-Abelian 1-form gauge theory with Dirac fields. The ordinary 4D non-Abelian theory, defined on the flat 4D Minkowski spacetime manifold, is generalized onto a (4, 2)-dimensional supermanifold which is parameterized by the spacetime bosonic coordinatesxμ(withμ=0,1,2,3) and a pair of Grassmannian variables (θ,θ-) which satisfy the standard relationships:θ2=θ-2=0and θθ-+θ-θ=0. Various consequences of the application of the above superspace (SUSP) unitary operator (and its Hermitian conjugate) are discussed. In particular, we obtain the results of the application of horizontality condition (HC) and gauge-invariant restriction (GIR) in the language of the above SUSP operators. One of the novel results of our present investigation is the derivation of explicit expressions for the SUSP unitary operator (and its Hermitian conjugate) without imposing any Hermitian conjugation condition fromoutsideon the parameters and (super)fields of the supersymmetric version of our 4D interacting non-Abelian 1-form theory with Dirac fields.

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Pressland, Matthew. "Calabi–Yau properties of Postnikov diagrams." Forum of Mathematics, Sigma 10 (2022). http://dx.doi.org/10.1017/fms.2022.52.

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Abstract We show that the dimer algebra of a connected Postnikov diagram in the disc is bimodule internally $3$ -Calabi–Yau in the sense of the author’s earlier work [43]. As a consequence, we obtain an additive categorification of the cluster algebra associated to the diagram, which (after inverting frozen variables) is isomorphic to the homogeneous coordinate ring of a positroid variety in the Grassmannian by a recent result of Galashin and Lam [18]. We show that our categorification can be realised as a full extension closed subcategory of Jensen–King–Su’s Grassmannian cluster category [28], in a way compatible with their bijection between rank $1$ modules and Plücker coordinates.

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Frieden, Gabriel. "Affine type A geometric crystal structure on the Grassmannian." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 28th... (April 22, 2020). http://dx.doi.org/10.46298/dmtcs.6393.

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International audience We construct a type A(1) n−1 affine geometric crystal structure on the Grassmannian Gr(k, n). The tropicalization of this structure recovers the combinatorics of crystal operators on semistandard Young tableaux of rectangular shape (with n − k rows), including the affine crystal operator e 0. In particular, the promotion operation on these tableaux essentially corresponds to cyclically shifting the Plu ̈cker coordinates of the Grassmannian.

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Bering, Klaus, and Michal Pazderka. "Symplectic Grassmannians, dual conformal symmetry and 4-point amplitudes in 6D." Journal of High Energy Physics 2022, no. 9 (September 6, 2022). http://dx.doi.org/10.1007/jhep09(2022)054.

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Abstract We investigate a new algebra-based approach of finding Grassmannian formulas for scattering amplitudes. Our prime motivation is massive amplitudes of 4D $$ \mathcal{N} $$ N = 4 SYM, and therefore we consider a 6D Grassmannian formula, where we can take advantage of massless kinematics. We next use symmetry arguments, and in particular, 6D dual conformal symmetry generalized to arbitrary dual conformal weights. Assuming a rational ansatz in terms of Plücker coordinates (i.e. minors) for the integrand, this approach leads to a set of algebraic equations. As an example, we explicitly find the solution for 4-point scattering amplitudes up to proportionality constants.

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MCMAHON, JORDAN, and NICHOLAS J. WILLIAMS. "THE COMBINATORICS OF TENSOR PRODUCTS OF HIGHER AUSLANDER ALGEBRAS OF TYPE A." Glasgow Mathematical Journal, July 29, 2020, 1–21. http://dx.doi.org/10.1017/s0017089520000361.

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Abstract We consider maximal non-l-intertwining collections, which are a higher-dimensional version of the maximal non-crossing collections which give clusters of Plücker coordinates in the Grassmannian coordinate ring, as described by Scott. We extend a method of Scott for producing such collections, which are related to tensor products of higher Auslander algebras of type A. We show that a higher preprojective algebra of the tensor product of two d-representation-finite algebras has a d-precluster-tilting subcategory. Finally, we relate mutations of these collections to a form of tilting for these algebras.

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29

Talaska, Kelli. "Combinatorial formulas for ⅃-coordinates in a totally nonnegative Grassmannian, extended abstract, extended abstract." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AK,..., Proceedings (January 1, 2009). http://dx.doi.org/10.46298/dmtcs.2706.

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International audience Postnikov constructed a decomposition of a totally nonnegative Grassmannian $(Gr _{kn})_≥0$ into positroid cells. We provide combinatorial formulas that allow one to decide which cell a given point in $(Gr _{kn})_≥0$ belongs to and to determine affine coordinates of the point within this cell. This simplifies Postnikov's description of the inverse boundary measurement map and generalizes formulas for the top cell given by Speyer and Williams. In addition, we identify a particular subset of Plücker coordinates as a totally positive base for the set of non-vanishing Plücker coordinates for a given positroid cell.

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30

Łukowski, Tomasz, Matteo Parisi, and Lauren K. Williams. "The Positive Tropical Grassmannian, the Hypersimplex, and the m = 2 Amplituhedron." International Mathematics Research Notices, March 7, 2023. http://dx.doi.org/10.1093/imrn/rnad010.

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Abstract The positive Grassmannian $Gr^{\geq 0}_{k,n}$ is a cell complex consisting of all points in the real Grassmannian whose Plücker coordinates are non-negative. In this paper we consider the image of the positive Grassmannian and its positroid cells under two different maps: the moment map$\mu $ onto the hypersimplex [ 31] and the amplituhedron map$\tilde{Z}$ onto the amplituhedron [ 6]. For either map, we define a positroid dissection to be a collection of images of positroid cells that are disjoint and cover a dense subset of the image. Positroid dissections of the hypersimplex are of interest because they include many matroid subdivisions; meanwhile, positroid dissections of the amplituhedron can be used to calculate the amplituhedron’s ‘volume’, which in turn computes scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills. We define a map we call T-duality from cells of $Gr^{\geq 0}_{k+1,n}$ to cells of $Gr^{\geq 0}_{k,n}$ and conjecture that it induces a bijection from positroid dissections of the hypersimplex $\Delta _{k+1,n}$ to positroid dissections of the amplituhedron $\mathcal{A}_{n,k,2}$; we prove this conjecture for the (infinite) class of BCFW dissections. We note that T-duality is particularly striking because the hypersimplex is an $(n-1)$-dimensional polytope while the amplituhedron $\mathcal{A}_{n,k,2}$ is a $2k$-dimensional non-polytopal subset of the Grassmannian $Gr_{k,k+2}$. Moreover, we prove that the positive tropical Grassmannian is the secondary fan for the regular positroid subdivisions of the hypersimplex, and prove that a matroid polytope is a positroid polytope if and only if all 2D faces are positroid polytopes. Finally, toward the goal of generalizing T-duality for higher $m$, we define the momentum amplituhedron for any even $m$.

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31

Karp, Steven N. "Sign variation, the Grassmannian, and total positivity." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 27th..., Proceedings (January 1, 2015). http://dx.doi.org/10.46298/dmtcs.2518.

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International audience The <i>totally nonnegative Grassmannian</i> is the set of $k$-dimensional subspaces $V$ of ℝ<sup>$n$</sup> whose nonzero Plücker coordinates (i.e. $k × k$ minors of a $k × n$ matrix whose rows span $V$) all have the same sign. Total positivity has been much studied in the past two decades from an algebraic, combinatorial, and topological perspective, but first arose in the theory of oscillations in analysis. It was in the latter context that Gantmakher and Krein (1950) and Schoenberg and Whitney (1951) independently showed that a subspace $V$ is totally nonnegative iff every vector in $V$, when viewed as a sequence of $n$ numbers and ignoring any zeros, changes sign fewer than $k$ times. We generalize this result, showing that the vectors in $V$ change sign fewer than $l$ times iff certain sequences of the Plücker coordinates of some <i>generic perturbation</i> of $V$ change sign fewer than $l − k + 1$ times. We give an algorithm which constructs such a generic perturbation. Also, we determine the <i>positroid cell</i> of each totally nonnegative $V$ from sign patterns of vectors in $V$. These results generalize to oriented matroids. La <i>grassmannienne totalement non négative</i> est l’ensemble des sous-espaces $V$ de ℝ<sup>$n$</sup> de dimension $k$ dont coordonnées plückeriennes non nulles (mineurs de l’ordre $k$ d’une matrice $k × n$ dont les lignes engendrent $V$) ont toutes le même signe. La positivité totale a beaucoup été étudiée durant les deux dernières décennies d’une perspective algébrique, combinatoire, et topologique, mais a pris naissance dans la théorie analytique des oscillations. C’est dans ce contexte que Gantmakher et Krein (1950) et Schoenberg et Whitney (1951) ont indépendamment démontré qu’un sous-espace $V$ est totalement non négatif ssi chaque vecteur dans $V$, lorsque considéré comme une séquence de $n$ nombres et dont on ignore les zéros, change de signe moins de $k$ fois. Nous généralisons ce résultat, démontrant que les vecteurs dans $V$ changent de signe moins de $l$ fois ssi certaines séquences des coordonnées plückeriennes d’une <i>perturbation générique</i> de $V$ changent de signe moins de $l − k + 1$ fois. Un algorithme construisant une telle perturbation générique est obtenu. De plus, nous déterminons la <i>cellule positroïde</i> de chaque $V$ totalement non négatif à partir des données de signe des vecteurs dans $V$. Ces résultats sont valides pour les matroïdes orientés.

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32

Sharma, Atul. "Ambidextrous light transforms for celestial amplitudes." Journal of High Energy Physics 2022, no. 1 (January 2022). http://dx.doi.org/10.1007/jhep01(2022)031.

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Abstract Low multiplicity celestial amplitudes of gluons and gravitons tend to be distributional in the celestial coordinates z,$$ \overline{z} $$ z ¯ . We provide a new systematic remedy to this situation by studying celestial amplitudes in a basis of light transformed boost eigenstates. Motivated by a novel equivalence between light transforms and Witten’s half-Fourier transforms to twistor space, we light transform every positive helicity state in the coordinate z and every negative helicity state in $$ \overline{z} $$ z ¯ . With examples, we show that this “ambidextrous” prescription beautifully recasts two- and three-point celestial amplitudes in terms of standard conformally covariant structures. These are used to extract examples of celestial OPE for light transformed operators. We also study such amplitudes at higher multiplicity by constructing the Grassmannian representation of tree-level gluon celestial amplitudes as well as their light transforms. The formulae for n-point Nk−2MHV amplitudes take the form of Euler-type integrals over regions in Gr(k, n) cut out by positive energy constraints.

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33

He, Song, and Zhenjie Li. "A note on letters of Yangian invariants." Journal of High Energy Physics 2021, no. 2 (February 2021). http://dx.doi.org/10.1007/jhep02(2021)155.

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Abstract Motivated by reformulating Yangian invariants in planar $$ \mathcal{N} $$ N = 4 SYM directly as d log forms on momentum-twistor space, we propose a purely algebraic problem of determining the arguments of the d log’s, which we call “letters”, for any Yangian invariant. These are functions of momentum twistors Z ’s, given by the positive coordinates α’s of parametrizations of the matrix C(α), evaluated on the support of polynomial equations C(α) · Z = 0. We provide evidence that the letters of Yangian invariants are related to the cluster algebra of Grassmannian G(4, n), which is relevant for the symbol alphabet of n-point scattering amplitudes. For n = 6, 7, the collection of letters for all Yangian invariants contains the cluster $$ \mathcal{A} $$ A coordinates of G(4, n). We determine algebraic letters of Yangian invariant associated with any “four-mass” box, which for n = 8 reproduce the 18 multiplicative-independent, algebraic symbol letters discovered recently for two-loop amplitudes.

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34

Jorge-Diaz, Carmen, Sabrina Pasterski, and Atul Sharma. "Celestial amplitudes in an ambidextrous basis." Journal of High Energy Physics 2023, no. 2 (February 15, 2023). http://dx.doi.org/10.1007/jhep02(2023)155.

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Abstract We start by constructing a conformally covariant improvement of the celestial light transform which keeps track of the mixing between incoming and outgoing states under finite Lorentz transformations in ℝ2,2. We then compute generic 2, 3 and 4-point celestial amplitudes for massless external states in the ambidextrous basis prepared by composing this SL(2, ℝ) intertwiner with the usual celestial map between momentum and boost eigenstates. The results are non-distributional in the celestial coordinates (z,$$ \overline{z} $$ z ¯ ) and conformally covariant in all scattering channels. Finally, we focus on the tree level 4-gluon amplitude where we present a streamlined route to the ambidextrous correlator based on Grassmannian formulae and examine its alpha space representation. In the process, we gain insights into the operator dictionary and CFT data of the holographic dual.

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35

Kumar, S., B. Chauhan, A. Tripathi, and R. P. Malik. "Massive 4D Abelian 2-form theory: Nilpotent symmetries from the (anti-)chiral superfield approach." International Journal of Modern Physics A 37, no. 02 (January 12, 2022). http://dx.doi.org/10.1142/s0217751x22500038.

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The off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetries are obtained by using the (anti-)chiral superfield approach (ACSA) to Becchi–Rouet–Stora–Tyutin (BRST) formalism for the four [Formula: see text]-dimensional (4D) Stückelberg-modified massive Abelian 2-form gauge theory. We perform exactly similar kind of exercise for the derivation of the off-shell nilpotent (anti-)co-BRST symmetry transformations, too. In the above derivations, the symmetry invariant restrictions on the superfields play very important and decisive roles. To prove the sanctity of the above nilpotent symmetries, we generalize our 4D ordinary theory (defined on the 4D flat Minkowskian space–time manifold) to its counterparts [Formula: see text]-dimensional (anti-)chiral super-submanifolds of the [Formula: see text]-dimensional supermanifold which is parametrized by the superspace coordinates [Formula: see text] where [Formula: see text] [Formula: see text] are the bosonic coordinates and a pair of Grassmannian variables [Formula: see text] are fermionic: [Formula: see text], [Formula: see text] in nature. One of the novel observations of our present endeavor is the derivation of the Curci–Ferrari (CF)-type restrictions from the requirement of the symmetry invariance of the coupled (but equivalent) Lagrangian densities of our theory within the framework of ACSA to BRST formalism. We also exploit the standard techniques of ACSA to capture the off-shell nilpotency and absolute anticommutativity of the conserved (anti-)BRST as well as the conserved (anti-)co-BRST charges. In a subtle manner, the proof of the absolute anticommutativity of the above conserved charges also implies the existence of the appropriate CF-type restrictions on our theory. This proof is also a novel result.

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36

Karp, Steven N. "Defining amplituhedra and Grassmann polytopes." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 28th... (April 22, 2020). http://dx.doi.org/10.46298/dmtcs.6356.

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International audience The totally nonnegative Grassmannian Gr≥0 k,n is the set of k-dimensional subspaces V of Rn whose nonzero Plucker coordinates all have the same sign. In their study of scattering amplitudes in N = 4 supersym- metric Yang-Mills theory, Arkani-Hamed and Trnka (2013) considered the image (called an amplituhedron) of Gr≥0 k,n under a linear map Z : Rn → Rr, where k ≤ r and the r × r minors of Z are all positive. One reason they required this positivity condition is to ensure that the map Gr≥0 k,n → Grk,r induced by Z is well defined, i.e. it takes everynelement of Gr≥0 k,n to a k-dimensional subspace of Rr. Lam (2015) gave a sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in which case he called the image a Grassmann polytope. (In the case k = 1, Grassmann polytopes are just polytopes, and amplituhedra are cyclic polytopes.) We give a necessary and sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in terms of sign variation. Using previous work we presented at FPSAC 2015, we obtain an equivalent condition in terms of the r × r minors of Z (assuming Z has rank r).

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37

Knutson, Allen, and Mathias Lederer. "Interval positroid varieties and a deformation of the ring of symmetric functions." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AT,..., Proceedings (January 1, 2014). http://dx.doi.org/10.46298/dmtcs.2453.

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International audience Define the <b>interval rank</b> $r_[i,j] : Gr_k(\mathbb C^n) →\mathbb{N}$ of a k-plane V as the dimension of the orthogonal projection $π _[i,j](V)$ of V to the $(j-i+1)$-dimensional subspace that uses the coordinates $i,i+1,\ldots,j$. By measuring all these ranks, we define the <b>interval rank stratification</b> of the Grassmannian $Gr_k(\mathbb C^n)$. It is finer than the Schubert and Richardson stratifications, and coarser than the positroid stratification studied by Lusztig, Postnikov, and others, so we call the closures of these strata <b>interval positroid varieties</b>. We connect Vakil's "geometric Littlewood-Richardson rule", in which he computed the homology classes of Richardson varieties (Schubert varieties intersected with opposite Schubert varieties), to Erd&odblac;s-Ko-Rado shifting, and show that all of Vakil's varieties are interval positroid varieties. We build on his work in three ways: (1) we extend it to arbitrary interval positroid varieties, (2) we use it to compute in equivariant K-theory, not just homology, and (3) we simplify Vakil's (2+1)-dimensional "checker games" to 2-dimensional diagrams we call "IP pipe dreams". The ring Symm of symmetric functions and its basis of Schur functions is well-known to be very closely related to the ring $\bigoplus_a,b H_*(Gr_a(\mathbb{C}^{(a+b)})$ and its basis of Schubert classes. We extend the latter ring to equivariant K-theory (with respect to a circle action on each $\mathbb{C}^{(a+b)}$, and compute the structure constants of this two-parameter deformation of Symm using the interval positroid technology above.

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38

Cotti, Giordano. "Coalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution of Prime Numbers." International Mathematics Research Notices, July 3, 2020. http://dx.doi.org/10.1093/imrn/rnaa163.

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Abstract The occurrence and frequency of a phenomenon of resonance (namely the coalescence of some Dubrovin canonical coordinates) in the locus of small quantum cohomology of complex Grassmannians are studied. It is shown that surprisingly this frequency is strictly subordinate and highly influenced by the distribution of prime numbers. Two equivalent formulations of the Riemann hypothesis are given in terms of numbers of complex Grassmannians without coalescence: the former as a constraint on the disposition of singularities of the analytic continuation of the Dirichlet series associated to the sequence counting non-coalescing Grassmannians, and the latter as asymptotic estimate (whose error term cannot be improved) for their distribution function.

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39

SPACEK, PETER. "LAURENT POLYNOMIAL LANDAU–GINZBURG MODELS FOR COMINUSCULE HOMOGENEOUS SPACES." Transformation Groups, January 27, 2021. http://dx.doi.org/10.1007/s00031-020-09636-7.

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AbstractIn this article we construct Laurent polynomial Landau–Ginzburg models for cominuscule homogeneous spaces. These Laurent polynomial potentials are defined on a particular algebraic torus inside the Lie-theoretic mirror model constructed for arbitrary homogeneous spaces in [Rie08]. The Laurent polynomial takes a similar shape to the one given in [Giv96] for projective complete intersections, i.e., it is the sum of the toric coordinates plus a quantum term. We also give a general enumeration method for the summands in the quantum term of the potential in terms of the quiver introduced in [CMP08], associated to the Langlands dual homogeneous space. This enumeration method generalizes the use of Young diagrams for Grassmannians and Lagrangian Grassmannians and can be defined type-independently. The obtained Laurent polynomials coincide with the results obtained so far in [PRW16] and [PR13] for quadrics and Lagrangian Grassmannians. We also obtain new Laurent polynomial Landau–Ginzburg models for orthogonal Grassmannians, the Cayley plane and the Freudenthal variety.

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40

Lam, Thomas. "Cyclic Demazure Modules and Positroid Varieties." Electronic Journal of Combinatorics 26, no. 2 (May 17, 2019). http://dx.doi.org/10.37236/8383.

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A positroid variety is an intersection of cyclically rotated Grassmannian Schubert varieties. Each graded piece of the homogeneous coordinate ring of a positroid variety is the intersection of cyclically rotated (rectangular) Demazure modules, which we call the cyclic Demazure module. In this note, we show that the cyclic Demazure module has a canonical basis, and define the cyclic Demazure crystal.

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41

Li, Li, James Mixco, B. Ransingh, and Ashish K. Srivastava. "An Introduction to Supersymmetric Cluster Algebras." Electronic Journal of Combinatorics 28, no. 1 (February 12, 2021). http://dx.doi.org/10.37236/9442.

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In this paper we propose the notion of cluster superalgebras which is a supersymmetric version of the classical cluster algebras introduced by Fomin and Zelevinsky. We show that the symplectic-orthogonal supergroup $SpO(2|1)$ admits a cluster superalgebra structure and as a consequence of this, we deduce that the supercommutative superalgebra generated by all the entries of a superfrieze is a subalgebra of a cluster superalgebra. We also show that the coordinate superalgebra of the super Grassmannian $G(2|0; 4|1)$ of chiral conformal superspace (that is, $(2|0)$ planes inside the superspace $\mathbb C^{4|1}$) is a quotient of a cluster superalgebra.

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42

Carotenuto, Alessandro, Fredy Díaz García, and Reamonn Ó Buachalla. "A Borel–Weil Theorem for the Irreducible Quantum Flag Manifolds." International Mathematics Research Notices, July 20, 2022. http://dx.doi.org/10.1093/imrn/rnac193.

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Abstract We establish a noncommutative generalisation of the Borel–Weil theorem for the Heckenberger–Kolb calculi of the irreducible quantum flag manifolds ${\mathcal {O}}_q(G/L_S)$, generalising previous work for the quantum Grassmannians ${\mathcal {O}}_q(\textrm {Gr}_{n,m})$. As a direct consequence we get a novel noncommutative differential geometric presentation of the quantum coordinate rings $S_q[G/L_S]$ of the irreducible quantum flag manifolds. The proof is formulated in terms of quantum principal bundles, and the recently introduced notion of a principal pair, and uses the Heckenberger and Kolb first-order differential calculus for the quantum Possion homogeneous spaces ${\mathcal {O}}_q(G/L^{\,\textrm {s}}_S)$.

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43

Farber, Miriam, and Pavel Galashin. "Weak Separation, Pure Domains and Cluster Distance." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 28th... (April 22, 2020). http://dx.doi.org/10.46298/dmtcs.6343.

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International audience Following the proof of the purity conjecture for weakly separated sets, recent years have revealed a variety of wider classes of pure domains in different settings. In this paper we show the purity for domains consisting of sets that are weakly separated from a pair of “generic” sets I and J. Our proof also gives a simple formula for the rank of these domains in terms of I and J. This is a new instance of the purity phenomenon which essentially differs from all previously known pure domains. We apply our result to calculate the cluster distance and to give lower bounds on the mutation distance between cluster variables in the cluster algebra structure on the coordinate ring of the Grassmannian. Using a linear projection that relates weak separation to the octahedron recurrence, we also find the exact mutation distances and cluster distances for a family of cluster variables.

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44

Bossinger, Lara. "Full-Rank Valuations and Toric Initial Ideals." International Mathematics Research Notices, April 13, 2020. http://dx.doi.org/10.1093/imrn/rnaa071.

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Abstract Let $V(I)$ be a polarized projective variety or a subvariety of a product of projective spaces, and let $A$ be its (multi-)homogeneous coordinate ring. To a full-rank valuation ${\mathfrak{v}}$ on $A$ we associate a weight vector $w_{\mathfrak{v}}$. Our main result is that the value semi-group of ${\mathfrak{v}}$ is generated by the images of the generators of $A$ if and only if the initial ideal of $I$ with respect to $w_{\mathfrak{v}}$ is prime. As application, we prove a conjecture by [ 7] connecting the Minkowski property of string polytopes to the tropical flag variety. For Rietsch-Williams’ valuation for Grassmannians, we identify a class of plabic graphs with non-integral associated Newton–Okounkov polytope (for ${\operatorname *{Gr}}_k(\mathbb C^n)$ with $n\ge 6$ and $k\ge 3$).

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45

Hochster, Melvin, Jack Jeffries, Vaibhav Pandey, and Anurag K. Singh. "When are the natural embeddings of classical invariant rings pure?" Forum of Mathematics, Sigma 11 (2023). http://dx.doi.org/10.1017/fms.2023.67.

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Abstract Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl’s book: For the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases, take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of S, equivalently, $S^G$ is a direct summand of S as an $S^G$ -module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular or the group G is linearly reductive.

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