Young Diagrams Research Articles

SYMMETRIC AND PSEUDO-SYMMETRIC NUMERICAL SEMIGROUPS VIA YOUNG DIAGRAMS AND THEIR SEMIGROUP RINGS

This paper studies Young diagrams of symmetric and pseudo-symmetric numerical semigroups and describes new operations on Young diagrams as well as numerical semigroups. These provide new decompositions of symmetric and pseudo-symmetric semigroups into an over semigroup and its dual. It is also given exactly for what kind of numerical semigroup $S$, the semigroup ring $\Bbbk[\![S]\!]$ has at least one Gorenstein subring and has at least one Kunz subring.

On Covering Numbers, Young Diagrams, and the Local Dimension of Posets

We study covering numbers and local covering numbers with respect to difference graphs and complete bipartite graphs. In particular, we show that in every cover of a Young diagram with $\binom{2k}{...

Stochastic dynamics of growing Young diagrams and their limit shapes

We investigate a class of Young diagrams growing via the addition of unit cells and satisfying the constraint that the height difference between adjacent columns ⩾r. In the long time limit, appropriately re-scaled Young diagrams approach a limit shape that we compute for each integer r ⩾ 0. We also determine limit shapes of ‘diffusively’ growing Young diagrams satisfying the same constraint and evolving through the addition and removal of cells that proceed with equal rates.

Eulerian polynomials via the Weyl algebra action

Through the action of the Weyl algebra on the geometric series, we establish a generalization of the Worpitzky identity and new recursive formulae for a family of polynomials including the classical Eulerian polynomials. We obtain an extension of the Dobiński formula for the sum of rook numbers of a Young diagram by replacing the geometric series with the exponential series. Also, by replacing the derivative operator with the q-derivative operator, we extend these results to the q-analogue setting including the q-hit numbers. Finally, a combinatorial description and a proof of the symmetry of a family of polynomials introduced by one of the authors are provided.

Magnificent four

We present a statistical mechanical model whose random variables are solid partitions, i.e. Young diagrams built by stacking up four dimensional hypercubes. Equivalently, it can be viewed as the model of random tessellations of \bf R^3 by squashed cubes of four fixed orientations. The model computes the refined index of a system of D0 -branes in the presence of D8–\overline{D8} system, with a B -field strong enough to support the bound states. Mathematically, it is the equivariant K-theoretic version of integration over the Hilbert scheme of points on \mathbb C^4 and its higher rank analogues, albeit the definition is real-, not complex analytic. The model is a mother of all random partition models, including the equivariant Donaldson–Thomas theory and the four dimensional instanton counting. Finally, a version of our model with infinite solid partitions with four fixed plane partition asymptotics is the vertex contribution to the equivariant count of instantons on toric Calabi–Yau fourfolds. The conjectured partition function of the model is presented. We have checked it up to six instantons (which is one step beyond the checks of the celebrated P. MacMahon's failed conjectures of the early XX century). A specialization of the formula is our earlier (2004) conjecture on the equivariant K-theoretic Donaldson–Thomas theory, recently proven by A. Okounkov [63].

Randomized Schützenberger’s Jeu De Taquin and Approximate Calculation of the Cotransition Probabilities of a Central Markov Process on the 3D Young Graph

There exists a well-known hook-length formula for calculating the dimensions of two-dimensional Young diagrams. Unfortunately, an analogous formula for the three-dimensional case is unknown. We introduce an approach to calculating estimated dimensions of three-dimensional Young diagrams, also known as plane partitions. The most difficult part of this task is the calculation of the cotransition probabilities for a central Markov process. We suggest an algorithm for an approximate calculation of these probabilities. It generates numerous random paths to a given diagram. In the case where the generated paths are uniformly distributed, the proportion of paths passing through a certain edge gives us an approximate value of the cotransition probability. As our numerical experiments show, a random generator based on the randomized version of the Schutzenberger transformation allows one to obtain accurate values of cotransition probabilities. Also, a method for constructing three-dimensional Young diagrams with large dimensions is suggested. Bibliography: 13 titles.

CRYSTAL STRUCTURES FOR SYMMETRIC GROTHENDIECK POLYNOMIALS

The symmetric Grothendieck polynomials representing Schubert classes in the $K$-theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type $A_n$ crystal structure on these tableaux. This crystal yields a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For single-columns and single-rows, we give a new combinatorial interpretation of Lascoux polynomials (K-analogs of Demazure characters) by constructing a K-theoretic analog of crystals with an appropriate analog of a Demazure crystal. We relate our crystal structure to combinatorial models using excited Young diagrams, Gelfand-Tsetlin patterns via the $5$-vertex model, and biwords via Hecke insertion to compute symmetric Grothendieck polynomials.

Optimization over Young diagrams

We consider the problem of finding a Young diagram minimizing the sum of evaluations of a given pair of functions on the parts of the associated pair of conjugate partitions. While there are exponentially many diagrams, we show it is polynomial time solvable.

Shiraishi functor and non-Kerov deformation of Macdonald polynomials

We suggest a further generalization of the hypergeometric-like series due to M. Noumi and J. Shiraishi by substituting the Pochhammer symbol with a nearly arbitrary function. Moreover, this generalization is valid for the entire Shiraishi series, not only for its Noumi–Shiraishi part. The theta function needed in the recently suggested description of the double-elliptic systems [Awata et al. JHEP 2020:150, arXiv:2005.10563, (2020)], 6d N = 2* SYM instanton calculus and the doubly-compactified network models, is a very particular member of this huge family. The series depends on two kinds of variables, vec {x} and vec {y}, and on a set of parameters, which becomes infinitely large now. Still, one of the parameters, p is distinguished by its role in the series grading. When vec {y} are restricted to a discrete subset labeled by Young diagrams, the series multiplied by a monomial factor reduces to a polynomial at any given order in p. All this makes the map from functions to the hypergeometric-like series very promising, and we call it Shiraishi functor despite it remains to be seen, what are exactly the morphisms that it preserves. Generalized Noumi–Shiraishi (GNS) symmetric polynomials inspired by the Shiraishi functor in the leading order in p can be obtained by a triangular transform from the Schur polynomials and possess an interesting grading. They provide a family of deformations of Macdonald polynomials, as rich as the family of Kerov functions, still very different from them, and, in fact, much closer to the Macdonald polynomials. In particular, unlike the Kerov case, these polynomials do not depend on the ordering of Young diagrams in the triangular expansion.

Affine Yangian and 3-Schur functions

3D (3 dimensional) Young diagram is a generalization of 2D Young diagram. In this paper, from the orthogonality of 3D Young diagrams and the properties in affine Yangian and its MacMahon representation, we obtain the Schur functions corresponding to 3D Young diagrams, which are called 3-Schur functions. 3-Schur functions are a generalization of Schur functions in the sense that when h1=1,h2=−1,h3=0, the 3-Schur functions of 3D Young diagrams become Schur functions of 2D Young diagrams, which is a special case of h1=h,h2=−1h,h3=1h−h. When h1=h,h2=−1h,h3=1h−h, the 3-Schur functions turn into the Jack symmetric polynomials of 2D Young diagrams by multiplying a coefficient. We will see that 3-Schur functions are symmetric about three coordinate axes.

Generalized $q,t$-Catalan numbers

Recent work of the first author, Negut and Rasmussen, and of Oblomkov and Rozansky in the context of Khovanov--Rozansky knot homology produces a family of polynomials in $q$ and $t$ labeled by integer sequences. These polynomials can be expressed as equivariant Euler characteristics of certain line bundles on flag Hilbert schemes. The $q,t$-Catalan numbers and their rational analogues are special cases of this construction. In this paper, we give a purely combinatorial treatment of these polynomials and show that in many cases they have nonnegative integer coefficients. For sequences of length at most 4, we prove that these coefficients enumerate subdiagrams in a certain fixed Young diagram and give an explicit symmetric chain decomposition of the set of such diagrams. This strengthens results of Lee, Li and Loehr for $(4,n)$ rational $q,t$-Catalan numbers.

Fock space dualities

A general theorem due to Howe of dual action of a classical group and a certain non-associative algebra on a space of symmetric or alternating tensors is reformulated in a setting of second quantization, and familiar examples in atomic and nuclear physics are discussed. The special case of orthogonal-orthogonal duality is treated in detail. It is shown that, like it was done by Helmers more than half a century ago in the analogous case of symplectic-symplectic duality, one can base a proof of the orthogonal-orthogonal duality theorem and a precise characterization of the relation between the equivalence classes of the dually related irreducible representations on a calculation of characters by combining it with an analysis of the representation of a reflection. Young diagrams for the description of equivalence classes of irreducible representations of orthogonal Lie algebras are introduced. The properties of a reflection of the number non-conserving part in the dual relationship between orthogonal Lie algebras corroborate a picture of an almost perfect symmetry between the partners.

Random permutations with logarithmic cycle weights

We consider random permutations on Sn with logarithmic growing cycles weights and study the asymptotic behavior as the length n tends to infinity. We show that the cycle count process converges to a vector of independent Poisson variables and also compute the total variation distance between both processes. Next, we prove a central limit theorem for the total number of cycles. Furthermore we establish a shape theorem and a functional central limit theorem for the Young diagrams associated to random permutations under this measure. We prove these results using tools from complex analysis and combinatorics. In particular we have to apply the method of singularity analysis to generating functions of the form exp((− log(1 − z))k+1) with k ≥ 1, which have not yet been studied in the literature.

Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagrams

We conjecture a closed-form expression of HOMFLY-PT invariants of double twist knots colored by rectangular Young diagrams where the twist is encoded in interpolation Macdonald polynomials. We also put forth a conjecture of cyclotomic expansions of HOMFLY-PT polynomials colored by rectangular Young diagrams for any knot.

AGT basis in SCFT for c = 3/2 and Uglov polynomials

AGT allows one to compute conformal blocks of d = 2 CFT for a large class of chiral CFT algebras. This is related to the existence of a certain orthogonal basis in the module of the (extended) chiral algebra. The elements of the basis are eigenvectors of a certain integrable model, labeled in general by N-tuples of Young diagrams. In particular, it was found that in the Virasoro case these vectors are expressed in terms of Jack polynomials, labeled by 2-tuples of ordinary Young diagrams, and for the super-Virasoro case they are related to Uglov polynomials, labeled by two colored Young diagrams. In the case of a generic central charge this statement was checked in the case when one of the Young diagrams is empty. In this note we study the N = 1 SCFT and construct 4 point correlation function using the basis. To this end we need to clarify the connection between basis elements and Uglov polynomials, we also need to use two bosonizations and their connection to the reflection operator. For the central charge c=3/2 we checked that there is a connection with the Uglov polynomials for the whole set of diagrams.

Double Jump Phase Transition in a Soliton Cellular Automaton

AbstractIn this paper, we consider the soliton cellular automaton introduced in [ 26] with a random initial configuration. We give multiple constructions of a Young diagram describing various statistics of the system in terms of familiar objects like birth-and-death chains and Galton–Watson forests. Using these ideas, we establish limit theorems showing that if the 1st $n$ boxes are occupied independently with probability $p\in (0,1)$, then the number of solitons is of order $n$ for all $p$ and the length of the longest soliton is of order $\log n$ for $p<1/2$, order $\sqrt{n}$ for $p=1/2$, and order $n$ for $p>1/2$. Additionally, we uncover a condensation phenomenon in the supercritical regime: for each fixed $j\geq 1$, the top $j$ soliton lengths have the same order as the longest for $p\leq 1/2$, whereas all but the longest have order $\log n$ for $p>1/2$. As an application, we obtain scaling limits for the lengths of the $k^{\textrm{th}}$ longest increasing and decreasing subsequences in a random stack-sortable permutation of length $n$ in terms of random walks and Brownian excursions.

Haldane Gap of the Three-Box Symmetric SU(3) Chain.

Motivated by the recent generalization of the Haldane conjecture to SU(3) chains [Lajkó etal., Nucl. Phys. B924, 508 (2017)NUPBBO0550-321310.1016/j.nuclphysb.2017.09.015] according to which a Haldane gap should be present for symmetric representations if the number of boxes in the Young diagram is a multiple of three, we develop a density matrix renormalization group algorithm based on standard Young tableaus to study the model with three boxes directly in the representations of the global SU(3) symmetry. We show that there is a finite gap between the singlet and the symmetric [3 0 0] sector Δ_{[3 0 0]}/J=0.040±0.006 where J is the antiferromagnetic Heisenberg coupling, and we argue on the basis of the structure of the low energy states that this is sufficient to conclude that the spectrum is gapped.

Lecture hall tableaux

We introduce lecture hall tableaux, which are fillings of a skew Young diagram satisfying certain conditions. Lecture hall tableaux generalize both lecture hall partitions and anti-lecture hall compositions, and also contain reverse semistandard Young tableaux as a limit case. We show that the coefficients in the Schur expansion of multivariate little q-Jacobi polynomials are generating functions for lecture hall tableaux. Using a Selberg-type integral we show that moments of multivariate little q-Jacobi polynomials, which are equal to generating functions for lecture hall tableaux of a Young diagram, have a product formula. We also explore various combinatorial properties of lecture hall tableaux.

Influence of resonances on the 11B(n,γ)12B capture reaction rate. Capture to the ground state of 12B

Within the framework of the modified potential cluster model with a classification of orbital states according to Young diagrams, the possibility of describing experimental data for total cross sections of the neutron radiative capture on 11B to the ground state of 12B at energies of 10 meV (1 meV = 10−3 eV) to 7 MeV was considered. It was shown that, taking into account only the E1 transition from the S state of the n11B scattering to the ground state of 12B, it is quite possible to explain the magnitude of the known experimental cross sections at energies of 25.3 meV to 70 keV. Furthermore, on the basis of the total cross sections of 10 meV to 7 MeV, but excluding resonances above 5 MeV, the reaction rate is calculated in the temperature range of 0.01 to 10.0 T9. It is shown that the inclusion of low-lying resonance states makes a significant contribution to the reaction rate, starting already with temperatures of 0.2–0.3 T9.

Representation stability of the cohomology of Springer varieties and some combinatorial consequences

A sequence of $$S_n$$ -representations $$\{V_n\}$$ is said to be uniformly representation stable if the decomposition of $$V_n = \bigoplus _{\mu } c_{\mu ,n} V(\mu )_n$$ into irreducible representations is independent of n for each $$\mu $$ —that is, the multiplicities $$c_{\mu ,n}$$ are eventually independent of n for each $$\mu $$ . Church–Ellenberg–Farb proved that the cohomology of flag varieties (the so-called diagonal coinvariant algebra) is uniformly representation stable. We generalize their result from flag varieties to all Springer fibers. More precisely, we show that for any increasing subsequence of Young diagrams, the corresponding sequence of Springer representations form a graded co-FI-module of finite type (in the sense of Church–Ellenberg–Farb). We also explore some combinatorial consequences of this stability.