Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams

We consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a q-analogue of permutations with restricted positions (i.e., rook placements). For general sets of entries, these numbers of matrices are not polynomials in q (Stembridge in Ann. Comb. 2(4):365, 1998); however, when the set of entries is a Young diagram, the numbers, up to a power of q?1, are polynomials with nonnegative coefficients (Haglund in Adv. Appl. Math. 20(4):450, 1998). In this paper, we give a number of conditions under which these numbers are polynomials in q, or even polynomials with nonnegative integer coefficients. We extend Haglund’s result to complements of skew Young diagrams, and we apply this result to the case where the set of entries is the Rothe diagram of a permutation. In particular, we give a necessary and sufficient condition on the permutation for its Rothe diagram to be the complement of a skew Young diagram up to rearrangement of rows and columns. We end by giving conjectures connecting invertible matrices whose support avoids a Rothe diagram and Poincaré polynomials of the strong Bruhat order.     

On hooks of Young diagrams

The well-known fact that there is always one more addable than removable box for a Young diagram is generalized to arbitrary hooks. As an application, this immediately implies a simple proof of a conjecture of Regev and Vershik [3] for which inductive proofs have recently been given by Regev and Zeilberger [4] and Janson [1].     

Enumeration of diagonally colored Young diagrams

This paper is concerned with the study of diffraction intensities of a relevant class of binary Pisot substitutions via exponential sums. Arithmetic properties of algebraic integers are used to give a new and constructive proof of the fact that there are no diffraction intensities outside the Fourier module of the underlying cut and project schemes. The results are then applied in the context of random substitutions.     

Bijections between pattern-avoiding fillings of Young diagrams

The pattern-avoiding fillings of Young diagrams we study arose from Postnikov's work on positive Grassmann cells. They are called -diagrams, and are in bijection with decorated permutations. Other closely-related fillings are interpreted as acyclic orientations of some bipartite graphs. The definition of the diagrams is the same but the avoided patterns are different. We give here bijections proving that the number of pattern-avoiding filling of a Young diagram is the same, for these two different sets of patterns. The result was obtained by Postnikov via a recurrence relation. This relation was extended by Spiridonov to obtain more general results about other patterns and other polyominoes than Young diagrams, and we show that our bijections also extend to more general polyominoes.     

A statistical sum associated with Young diagrams

We study the asymptotics of sums of powers of normalized dimensions of complex irreducible representations of the symmetric group as N. We calculate the limit Gibbs measure on the space of Young diagrams. The problem is connected with a one-dimensional model of statistical physics. In the appendix, written by A. M. Pass, numerical and graphical data about the free energy corresponding to the statistical studied are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 164, pp. 20–29, 1987.     

Limiting shapes of birth-and-death processes on Young diagrams

We consider a family of birth processes and birth-and-death processes on Young diagrams of integer partitions of n. This family incorporates three famous models from very different fields: Rost?s totally asymmetric particle model (in discrete time), Simon?s urban growth model, and Moran?s infinite alleles model. We study stationary distributions and limit shapes as n tends to infinity, and present a number of results and conjectures.     

Rational representations of Yangians associated with skew Young diagrams

Consider the general linear group GL_M over the complex field. The irreducible rational representations of the group GL_M can be labeled by the pairs of partitions and such that the total number of non-zero parts of and does not exceed M. Let EQ4 be the irreducible representation corresponding to such a pair. Regard the direct product as a subgroup of GL_N+M . Take any irreducible rational representation of GL_N+M. The vector space comes with a natural action of the group GL_N. Put n=. For any pair of standard Young tableaux of skew shapes respectively, we give a realization of as a subspace in the tensor product of n copies of defining representation of GL_N, and of ñ copies of the contragredient representation ()^*. This subspace is determined as the image of a certain linear operator on W_nñn. We introduce this operator by an explicit multiplicative formula. When M=0 and is an irreducible representation of GL_N, we recover the known realization of as a certain subspace in the space of all traceless tensors in . Then the operator may be regarded as the rational analogue of the Young symmetrizer, corresponding to the tableau of shape . Even when M=0, our formula for is new. Our results are applications of the representation theory of the Yangian of the Lie algebra . In particular, is an intertwining operator between certain representations of the algebra on . We also introduce the notion of a rational representation of the Yangian . As a representation of , the image of is rational and irreducible.Mathematics Subject Classification (2000): 17B37, 20C30, 22E46, 81R50in final form: 10 July 2003     

Representations of twisted Yangians associated with skew Young diagrams

Let $G_M$ be either the orthogonal group $O_M$ or the symplectic group $Sp_M$ over the complex field; in the latter case the non-negative integer $M$ has to be even. Classically, the irreducible polynomial representations of the group $G_M$ are labeled by partitions $\mu=(\mu_{1},\mu_{2},\,\ldots)$ such that $\mu^{\prime}_1+\mu^{\prime}_2\le M$ in the case $G_M=O_M$, or $2\mu^{\prime}_1\le M$ in the case $G_M=Sp_M$. Here $\mu^{\prime}=(\mu^{\prime}_{1},\mu^{\prime}_{2},\,\ldots)$ is the partition conjugate to $\mu$. Let $W_\mu$ be the irreducible polynomial representation of the group $G_M$ corresponding to $\mu$. Regard $G_N\times G_M$ as a subgroup of $G_{N+M}$. Then take any irreducible polynomial representation $W_\lambda$ of the group $G_{N+M}$. The vector space $W_{\lambda}(\mu)={\rm Hom}_{\,G_M}( W_\mu, W_\lambda)$ comes with a natural action of the group $G_N$. Put $n=\lambda_1-\mu_1+\lambda_2-\mu_2+\ldots\,$. In this article, for any standard Young tableau $\varOmega$ of skew shape $\lm$ we give a realization of $W_{\lambda}(\mu)$ as a subspace in the $n$-fold tensor product $(\mathbb{C}^N)^{\bigotimes n}$, compatible with the action of the group $G_N$. This subspace is determined as the image of a certain linear operator $F_\varOmega (M)$ on $(\mathbb{C}^N)^{\bigotimes n}$, given by an explicit formula. When $M=0$ and $W_{\lambda}(\mu)=W_\lambda$ is an irreducible representation of the group $G_N$, we recover the classical realization of $W_\lambda$ as a subspace in the space of all traceless tensors in $(\mathbb{C}^N)^{\bigotimes n}$. Then the operator $F_\varOmega\(0)$ may be regarded as the analogue for $G_N$ of the Young symmetrizer, corresponding to the standard tableau $\varOmega$ of shape $\lambda$. This symmetrizer is a certain linear operator on $\CNn$$(\mathbb{C}^N)^{\bigotimes n} $ with the image equivalent to the irreducible polynomial representation of the complex general linear group $GL_N$, corresponding to the partition $\lambda$. Even in the case $M=0$, our formula for the operator $F_\varOmega(M)$ is new. Our results are applications of the representation theory of the twisted Yangian, corresponding to the subgroup $G_N$ of $GL_N$. This twisted Yangian is a certain one-sided coideal subalgebra of the Yangian corresponding to $GL_N$. In particular, $F_\varOmega(M)$ is an intertwining operator between certain representations of the twisted Yangian in $(\mathbb{C}^N)^{\bigotimes n}$.     

Diophantine geometry over groups I: Makanin-Razborov diagrams

Manucsrit re?u le 4 janvier 2000.     

Projective diagrams over partially ordered sets are free

A generalization of Kaplansky's theorem on projective modules over local rings is used to show that every projective diagram over a poset is free.     

Gaussian fluctuations of characters of symmetric groups and of Young diagrams

We study asymptotics of reducible representations of the symmetric groups S _q for large q. We decompose such a representation as a sum of irreducible components (or, alternatively, Young diagrams) and we ask what is the character of a randomly chosen component (or, what is the shape of a randomly chosen Young diagram). Our main result is that for a large class of representations the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are asymptotically Gaussian; in this way we generalize Kerov's central limit theorem. The considered class consists of representations for which the characters almost factorize and this class includes, for example, the left-regular representation (Plancherel measure), irreducible representations and tensor representations. This class is also closed under induction, restriction, outer product and tensor product of representations. Our main tool in the proof is the method of genus expansion, well known from the random matrix theory.     

A bijective proof of the Jacobi identity and reconstruction of Young diagrams

A short combinatorial proof is given of the classical Jacobi identity from the theory of theta-functions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 155, pp. 3–6, 1986.     

Hall–Littlewood polynomials, alcove walks, and fillings of Young diagrams

A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of Young diagrams. The inversion statistic, which is the more intricate one, suffices for specializing a closely related formula to one for the type A Hall–Littlewood Q-polynomials (spherical functions on p-adic groups). An apparently unrelated development, at the level of arbitrary finite root systems, led to Schwer’s formula (rephrased and rederived by Ram) for the Hall–Littlewood P-polynomials of arbitrary type. The latter formula is in terms of so-called alcove walks, which originate in the work of Gaussent–Littelmann and of the author with Postnikov on discrete counterparts to the Littelmann path model. In this paper, we relate the above developments, by deriving a Haglund–Haiman–Loehr type formula for the Hall–Littlewood P-polynomials of type A from Ram’s version of Schwer’s formula via a “compression” procedure.