Algebra of differential operators associated with Young diagrams

Abstract

We establish a correspondence between Young diagrams and differential operators of infinitely many variables. These operators form a commutative associative algebra isomorphic to the algebra of the conjugated classes of finite permutations of the set of natural numbers. The Schur functions form a complete system of common eigenfunctions of these differential operators, and their eigenvalues are expressed through the characters of symmetric groups. The structure constants of the algebra are expressed through the Hurwitz numbers.

11 pages

References in corpus (4)

• Generation of Matrix Models by W-operators
• Virasoro constraints for Kontsevich-Hurwitz partition function
• The Algebra of Conjugacy Classes in Symmetric Groups and Partial Permutations
• Disk single Hurwitz numbers

Cited by in corpus (78)

• Superpolynomials for toric knots from evolution induced by cut-and-join operators
• On KP-integrable Hurwitz functions
• Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid
• Colored HOMFLY polynomials of knots presented as double fat diagrams
• On the complete perturbative solution of one-matrix models
• Correlators in tensor models from character calculus
• Ward identities and combinatorics of rainbow tensor models
• Colored HOMFLY polynomials for the pretzel knots and links
• Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for 3-strand knots in any symmetric and antisymmetric representations
• Character expansion for HOMFLY polynomials. I. Integrability and difference equations
• Superintegrability for ($\beta$-deformed) partition function hierarchies with $W$-representations
• On universal knot polynomials
• Superintegrability summary
• On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions
• Towards effective topological field theory for knots
• Toric Calabi-Yau threefolds as quantum integrable systems. R-matrix and RTT relations
• On colored HOMFLY polynomials for twist knots
• Tabulating knot polynomials for arborescent knots
• Knot polynomials in the first non-symmetric representation
• Knot invariants from Virasoro related representation and pretzel knots
• Sum rules for characters from character-preservation property of matrix models
• Cut and join operator ring in Aristotelian tensor model
• On KP-integrable skew Hurwitz $\tau$-functions and their $\beta$-deformations
• Towards matrix model representation of HOMFLY polynomials
• Equations on knot polynomials and 3d/5d duality
• Interplay between MacDonald and Hall-Littlewood expansions of extended torus superpolynomials
• Many-body integrable systems implied by WLZZ models
• On genus expansion of superpolynomials
• Torus HOMFLY as the Hall-Littlewood Polynomials
• Cut-and-join structure and integrability for spin Hurwitz numbers
• From Kronecker to tableau pseudo-characters in tensor models
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• CFT approach to constraint operators for ($\beta$-deformed) hermitian one-matrix models
• Around spin Hurwitz numbers
• Complete solution to Gaussian tensor model and its integrable properties
• Hopf link invariants and integrable hierarchies
• Integrability properties of Hurwitz partition functions. II. Multiplication of cut-and-join operators and WDVV equations
• $(q,t)$-deformed (skew) Hurwitz $\tau$-functions
• An analogue of Schur functions for the plane partitions
• Kerov functions revisited
• Commutative families in $W_{\infty }$, integrable many-body systems and hypergeometric $\tau$-functions
• Distinguishing Mutant Knots
• Bilinear character correlators in superintegrable theory
• Spectral curves and $W$-representations of matrix models
• On Products of Random Matrices
• Checks of integrality properties in topological strings
• Challenges of beta-deformation
• Tensorial generalization of characters
• Matching branches of non-perturbative conformal block at its singularity divisor
• A Hurwitz theory avatar of open-closed strings
• Colored Alexander polynomials and KP hierarchy
• Connection between cut-and-join and Casimir operators
• On the status of DELL systems
• Hurwitz numbers from Feynman diagrams
• On Hamiltonians for Kerov functions
• Chern-Simons perturbative series revisited
• A new symmetry of the colored Alexander polynomial
• Spin Hurwitz theory and Miwa transform for the Schur Q-functions
• Colored HOMFLY-PT polynomials of quasi-alternating $3$-braid knots
• Integrable hierarchies, Hurwitz numbers and a branch point field in critical topologically massive gravity
• On generalized Macdonald polynomials
• Cardy-Frobenius extension of algebra of cut-and-join operators
• Macdonald polynomials for super-partitions
• Chalykh's Baker-Akhiezer functions as eigenfunctions of the integer-ray integrable systems
• Hook variables: cut-and-join operators and $\tau$-functions
• Colored HOMFLY-PT for hybrid weaving knot ${\hat{W}}_3(m,n)$
• $W_{1+\infty }$ and $\tilde{W}$ algebras, and Ward identities
• Deformation of superintegrability in the Miwa-deformed Gaussian matrix model
• Asymptotic Hurwitz numbers
• From Hurwitz numbers to Feynman diagrams: counting rooted trees in log gravity
• Generalized $\tilde{W}$ algebras
• Matrix model and dimensions at hypercube vertices
• Averaging method in combinatorics of symmetric polynomials
• Genus Expanded Cut-and-Join operators and generalized Hurwtiz numbers
• A Formula about W-operator and Its Application to Hurwitz Number
• CMM formula as superintegrability property of unitary model
• Degree of W-operator and Noncrossing Partition
• Shifted genus expanded ${\mathcal{W}}_{\infty }$ algebra and shifted Hurwtiz numbers