Symmetry groups of pfaffians of symmetric matrices

Pfaffian L-ensembles related to the z-measures on partitions with the Jack parameters 1/2 or 2

We construct Pfaffian L-ensembles related to the z-measures on partitions and to the Plancherel measures on partitions with the Jack parameters 1/2 or 2. The results imply that these measures on partitions lead to Pfaffian point processes, and the correlation kernels of these processes can be expressed in terms of the corresponding L-matrices. We give explit formulae for these L-matrices.

Minors of a skew symmetric matrix: A combinatorial approach

We use Knuth's combinatorial approach to Pfaffians to reprove and clarify a century-old formula, due to Brill. It expresses arbitrary minors of a skew symmetric matrix in terms of Pfaffians.

A sign-reversing involution for an extension of Torelli's Pfaffian identity

We evaluate the hyperpfaffian of a skew-symmetric $k$-ary polynomial $f$ of degree $k/2 \cdot (n-1)$. The result is a product of the Vandermonde product and a certain expression involving the coefficients of the polynomial $f$. The proof utilizes a sign reversing involution on a set of weighted, oriented partitions. When restricting to the classical case when $k=2$ and the polynomial is $(x_{j} - x_{i})^{n-1}$, we obtain an identity due to Torelli.

Hankel Pfaffians, Discriminants and Kazhdan-Lusztig bases

We use Kazhdan-Lusztig bases of representations of the symmetric group to express Pfaffians with entries $(a_i-a_j) h_{i+j}$. In the case where the parameters $a_i$ are specialized to successive powers of $q$, and the $h_i$ are complete functions, we obtain the $q$-discriminant.

On refined enumerations of totally symmetric self-complementary plane partitions I

In this paper we give Pfaffian expressions and constant term identities for the enumeration problems presented by Mills, Robbins and Rumsey (``Self-complementary totally symmetric plane partitions'' J. Combin. Theory Ser. A, 42, 277--292), concerning the totally symmetric self-complementary plane partitions (TSSCPPs). We also present some new conjectures and give Pfaffian expressions for them.

Aztec菱形的反对角对称多米诺铺砌

Off-diagonally symmetric domino tilings of the Aztec diamond

本文引入了一种新的多米诺铺砌对称类，通过非相交格点路径和Stembridge的非相交格点路径Pfaffian公式变种，计算了Aztec菱形的反对角对称多米诺铺砌数量。

对角对称交替符号矩阵

Diagonally symmetric alternating sign matrices

本文研究对角对称交替符号矩阵（DSASMs）的计数，得到其数量及相关统计量生成函数的Pfaffian公式，引入与DSASM一一对应的六顶点模型变体并研究相关主题。

The z-measures on partitions, Pfaffian point processes, and the matrix hypergeometric kernel

We consider a point process on one-dimensional lattice originated from the harmonic analysis on the infinite symmetric group, and defined by the z-measures with the deformation (Jack) parameter 2. We derive an exact Pfaffian formula for the correlation function of this process. Namely, we prove that the correlation function is given as a Pfaffian with a matrix kernel. The kernel is given in terms of the Gauss hypergeometric functions, and can be considered as a matrix analogue of the Hypergeometric kernel introduced by A. Borodin and G. Olshanski. Our result holds for all values of admissible …

Four symmetry classes of plane partitions under one roof

In previous paper, the author applied the permanent-determinant method of Kasteleyn and its non-bipartite generalization, the Hafnian-Pfaffian method, to obtain a determinant or a Pfaffian that enumerates each of the ten symmetry classes of plane partitions. After a cosmetic generalization of the Kasteleyn method, we identify the matrices in the four determinantal cases (plain plane partitions, cyclically symmetric plane partitions, transpose-complement plane partitions, and the intersection of the last two types) in the representation theory of sl(2,C). The result is a unified proof of the fo…

The Faddeev-LeVerrier algorithm and the Pfaffian

We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost $O(n^{β+1})$ where $n$ is the size of the matrix and $O(n^β)$ is the cost of multiplying $n\times n$-matrices, $β\in[2,2.37286)$. We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold using computer algebra.

更多小项求和公式

More minor summation formulae

本文推导出同时推广Ishikawa和Wakayama以及Byun最近小项求和公式的行列式-佩尔数公式，基于斜对称矩阵与秩-1矩阵和的行列式分解公式，并应用于斜Schur函数的Cauchy型恒等式。

Symmetry classes of alternating-sign matrices under one roof

In a previous article [math.CO/9712207], we derived the alternating-sign matrix (ASM) theorem from the Izergin-Korepin determinant for a partition function for square ice with domain wall boundary. Here we show that the same argument enumerates three other symmetry classes of alternating-sign matrices: VSASMs (vertically symmetric ASMs), even HTSASMs (half-turn-symmetric ASMs), and even QTSASMs (quarter-turn-symmetric ASMs). The VSASM enumeration was conjectured by Mills; the others by Robbins [math.CO/0008045]. We introduce several new types of ASMs: UASMs (ASMs with a U-turn side), UUASMs (t…

论广义Pfaffian（普法夫式）

On Generalized Pfaffians

受S类超共形场论研究启发的“反对称矩阵元素的系列多项式存在多项式平方根”这一广义Pfaffian猜想，此前仅证实部分特例，本文完整证明了该猜想，其推论含D型局部Hitchin像的刻画。

P-Grothendieck多项式的次数与Pfaffian簇的正则性

Degrees of P-Grothendieck polynomials and regularity of Pfaffian varieties

本研究利用移位杨表组合数学证明P-Grothendieck多项式次数公式，结合已有结果得到vexillary斜对称矩阵Schubert簇的Castelnuovo-Mumford正则性上界，还得到Grassmannian相关多项式次数与簇正则性的新公式。

Symmetric, Hankel-symmetric, and Centrosymmetric Doubly Stochastic Matrices

We investigate convex polytopes of doubly stochastic matrices having special structures: symmetric, Hankel symmetric, centrosymmetric, and both symmetric and Hankel symmetric. We determine dimensions of these polytopes and classify their extreme points. We also determine a basis of the real vector spaces generated by permutation matrices with these special structures.

Symmetric and skew-symmetric $\{0,\pm 1\}$-matrices with large determinants

We show that the existence of $\{\pm 1\}$-matrices having largest possible determinant is equivalent to the existence of certain tournament matrices. In particular, we prove a recent conjecture of Armario. We also show that large submatrices of conference matrices are determined by their spectrum.

The Propus Construction for Symmetric Hadamard Matrices

\textit{Propus} (which means twins) is a construction method for orthogonal $\pm 1$ matrices based on a variation of the Williamson array called the \textit{propus array} \[ \begin{matrix*}[r] A& B & B & D B& D & -A &-B B& -A & -D & B D& -B & B &-A. \end{matrix*} \] This construction designed to find symmetric Hadamard matrices was originally based on circulant symmetric $\pm 1$ matrices, called \textit{propus matrices}. We also give another construction based on symmetric Williamson-type matrices. We give constructions to find symmetric propus-Hadamard matrices for 57 orders $4n$, $n < 200$ o…

Involutions of the Symmetric Group and Congruence B-orbits of Anti-Symmetric Matrices

We present the poset of Borel congruence classes of anti-symmetric matrices ordered by containment of closures. We show that there exists a bijection between the set of these classes and the set of involutions of the symmetric group. We give two formulas for the rank function of this poset.

Symmetric Bush-type Hadamard matrices of order $4m^4$ exist for all odd $m$

Using reversible Hadamard difference sets, we construct symmetric Bush-type Hadamard matrices of order $4m^4$ for all odd integer $m$.

Multivariable Christoffel-Darboux Kernels and Characteristic Polynomials of Random Hermitian Matrices

We study multivariable Christoffel-Darboux kernels, which may be viewed as reproducing kernels for antisymmetric orthogonal polynomials, and also as correlation functions for products of characteristic polynomials of random Hermitian matrices. Using their interpretation as reproducing kernels, we obtain simple proofs of Pfaffian and determinant formulas, as well as Schur polynomial expansions, for such kernels. In subsequent work, these results are applied in combinatorics (enumeration of marked shifted tableaux) and number theory (representation of integers as sums of squares).

Integer symmetric matrices of small spectral radius and small Mahler measure

In a previous paper we completely described cyclotomic matrices--integer symmetric matrices of spectral radius at most 2. In this paper we find all minimal noncyclotomic matrices. As a consequence, we are able to determine all integer symmetric matrices of spectral radius at most 2.019, and to determine all integer symmetric matrices whose Mahler measure is at most 1.3. In particular we solve the strong version of Lehmer's problem for integer symmetric matrices: all noncyclotomic matrices have Mahler measure at least "Lehmer's number" 1.17628... .

The Group Inverse of extended Symmetric and Periodic Jacobi Matrices

In this work, we explicitly compute the group inverse of symmetric and periodic Jacobi matrices.

Product Matrix Processes with Symplectic and Orthogonal Invariance via Symmetric Functions

We apply symmetric function theory to study random processes formed by singular values of products of truncations of Haar distributed symplectic and orthogonal matrices. These product matrix processes are degenerations of Macdonald processes introduced by Borodin and Corwin. Through this connection, we obtain explicit formulae for the distribution of singular values of a deterministic matrix multiplied by a truncated Haar orthogonal or symplectic matrix under conditions where the latter factor acts as a rank $1$ perturbation. Consequently, we generalize the recent Kieburg-Kuijlaars-Stivigny fo…