The geometric Burge correspondence and the partition function of polymer replicas

We construct a geometric lifting of the Burge correspondence as a composition of local birational maps on generic Young-diagram-shaped arrays. We establish its fundamental relation to the geometric Robinson-Schensted-Knuth correspondence and to the geometric Schützenberger involution. We also show a number of properties of the geometric Burge correspondence, specializing them to the case of symmetric input arrays. In particular, our construction shows that such a mapping is volume preserving in log-log variables. As an application, we consider a model of two polymer paths of given length constrained to have the same endpoint, known as polymer replica. We prove that the distribution of the polymer replica partition function in a log-gamma random environment is a Whittaker measure, and deduce the corresponding Whittaker integral identity. For a certain choice of the parameters, we notice a distributional identity between our model and the symmetric log-gamma polymer studied by O’Connell, Seppäläinen, and Zygouras (2014).

Revisiting higher-spin gyromagnetic couplings

We analyze Bosonic, Heterotic, and Type II string theories compactified on a generic torus having constant moduli. By computing the hamiltonian giving the interaction between massive string excitations and U(1) gauge fields arising from the graviton and Kalb-Ramond field upon compactification, we derive a general formula for such couplings that turns out to be universal in all these theories. We also confirm our result by explicitly evaluating the relevant string three-point amplitudes. From this expression, we determine the gyromagnetic ratio g of massive string states coupled to both gauge-fields. For a generic mixed symmetry state, there is one gyromagnetic coupling associated with each row of the corresponding Young Tableau diagram. For all the states having zero Kaluza Klein or Winding charges, the value of g turns out to be 1. We also explicitly consider totally symmetric and mixed symmetry states (having two rows in the Young diagram) associated with the first Regge-trajectory and obtain their corresponding g value.

Duality in elliptic Ruijsenaars system and elliptic symmetric functions

We demonstrate that the symmetric elliptic polynomials $$E_\lambda (x)$$ E λ ( x ) originally discovered in the study of generalized Noumi–Shiraishi functions are eigenfunctions of the elliptic Ruijsenaars–Schneider (eRS) Hamiltonians that act on the mother function variable $$y_i$$ y i (substitute of the Young-diagram variable $$\lambda $$ λ ). This means they are eigenfunctions of the dual eRS system. At the same time, their orthogonal complements in the Schur scalar product, $$P_\lambda (x)$$ P λ ( x ) are eigenfunctions of the elliptic reduction of the Koroteev–Shakirov (KS) Hamiltonians. This means that these latter are related to the dual eRS Hamiltonians by a somewhat mysterious orthogonality transformation, which is well defined only on the full space of time variables, while the coordinates $$x_i$$ x i appear only after the Miwa transform. This observation explains the difficulties with getting the apparently self-dual Hamiltonians from the double elliptic version of the KS Hamiltonians.

Quarter-BPS states, multi-symmetric functions and set partitions

We give a construction of general holomorphic quarter BPS operators in $$ \mathcal{N} $$ N = 4 SYM at weak coupling with U(N) gauge group at finite N. The construction employs the Möbius inversion formula for set partitions, applied to multi-symmetric functions, alongside computations in the group algebras of symmetric groups. We present a computational algorithm which produces an orthogonal basis for the physical inner product on the space of holomorphic operators. The basis is labelled by a U(2) Young diagram, a U(N) Young diagram and an additional plethystic multiplicity label. We describe precision counting results of quarter BPS states which are expected to be reproducible from dual computations with giant gravitons in the bulk, including a symmetry relating sphere and AdS giants within the quarter BPS sector. In the case n ≤ N (n being the dimension of the composite operator) the construction is analytic, using multi-symmetric functions and U(2) Clebsch-Gordan coefficients. Counting and correlators of the BPS operators can be encoded in a two-dimensional topological field theory based on permutation algebras and equipped with appropriate defects.

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.

Double Jump Phase Transition in a Soliton Cellular Automaton

In 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.

A combinatorial approach to Macdonald q, t-symmetry via the Carlitz bijection

International audience We investigate the combinatorics of the symmetry relation H μ(x; q, t) = H μ∗ (x; t, q) on the transformed Macdonald polynomials, from the point of view of the combinatorial formula of Haglund, Haiman, and Loehr in terms of the inv and maj statistics on Young diagram fillings. By generalizing the Carlitz bijection on permutations, we provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q = 0) for the coefficients of the square-free monomials in the variables x. Our work in this case relates the Macdonald inv and maj statistics to the monomial basis of the modules Rμ studied by Garsia and Procesi. We also provide a new proof for the full Macdonald relation in the case when μ is a hook shape.

Search for Young diagrams with large dimensions

Search for Young diagrams with maximum dimensions or, equivalently, search for irreducible representations of the symmetric group $S(n)$ with maximum dimensions is an important problem of asymptotic combinatorics. In this paper, we propose algorithms that transform a Young diagram into another one of the same size but with a larger dimension. As a result of massive numerical experiments, the sequence of $10^6$ Young diagrams with large dimensions was constructed. Furthermore, the proposed algorithms do not change the first 1000 elements of this sequence. This may indicate that most of them have the maximum dimension. It has been found that the dimensions of all Young diagrams of the resulting sequence starting from the 75778th exceed the dimensions of corresponding diagrams of the greedy Plancherel sequence.

Crystals, Semistandard Tableaux and Cyclic Sieving Phenomenon

In this paper, we study a new cyclic sieving phenomenon on the set $\mathsf{SST}_n(\lambda)$ of semistandard Young tableaux with the cyclic action $\mathsf{c}$ arising from its $U_q(\mathfrak{sl}_n)$-crystal structure. We prove that if $\lambda$ is a Young diagram with $\ell(\lambda) < n$ and $\gcd( n, |\lambda| )=1$, then the triple $\left( \mathsf{SST}_n(\lambda), \mathsf{C}, q^{- \kappa(\lambda)} s_\lambda(1,q, \ldots, q^{n-1}) \right) $ exhibits the cyclic sieving phenomenon, where $\mathsf{C}$ is the cyclic group generated by $\mathsf{c}$. We further investigate a connection between $\mathsf{c}$ and the promotion $\mathsf{pr}$ and show the bicyclic sieving phenomenon given by $\mathsf{c}$ and $\mathsf{pr}^n$ for hook shape.