相关论文: Combinatoric of H-primes in quantum matrices

Let $k$ be a nonnegative integer, and let $\alpha$ and $\beta$ be two permutations of $n$ symbols. We say that $\alpha$ and $\beta$ $k$-commute if $H(\alpha\beta, \beta\alpha)=k$, where $H$ denotes the Hamming metric between permutations.…

In this note we construct a poset map from a Boolean algebra to the Bruhat order which unveils an interesting connection between subword complexes, sorting orders, and certain totally nonnegative spaces. This relationship gives a new proof…

Let (G,H) be one of the equal rank reductive dual pairs (Mp_{2n},O_{2n+1}) or (U_n,U_n) over a non-archimedean local field of characteristic zero. It is well-known that the theta correspondence establishes a bijection between certain…

In a quantum computer, creating superpositions of quantum bits (qubits) in different states can lead to a speed-up over classical computers [1], but quantum mechanics also allows for the superposition of quantum circuits [2]. In fact, it…

Let $s$ be West's deterministic stack-sorting map. A well-known result (West) is that any length $n$ permutation can be sorted with $n-1$ iterations of $s.$ In 2020, Defant introduced the notion of highly-sorted permutations -- permutations…

Let $I_n$ be the set of involutions in the symmetric group $S_n$, and for $A \subseteq \{0,1,\ldots,n\}$, let \[ F_n^A=\{\sigma \in I_n \mid \text{$\sigma$ has $a$ fixed points for some $a \in A$}\}. \] We give a complete characterisation…

Given an n by n matrix A over the complex numbers and an invariant subspace L, this paper gives a straightforward formula to construct an n by n matrix N that commutes with A and has L equal to the kernel of N. For Q a matrix putting A into…

The intrinsic symmetries of physical systems have been employed to reduce the number of degrees of freedom of systems, thereby simplifying computations. In this work, we investigate the properties of $\mathcal{M}SU(2^N)$,…

We show that any smooth permutation $\sigma\in S_n$ is characterized by the set ${\mathbf{C}}(\sigma)$ of transpositions and $3$-cycles in the Bruhat interval $(S_n)_{\leq\sigma}$, and that $\sigma$ is the product (in a certain order) of…

Motivated in part by a problem of combinatorial optimization and in part by analogies with quantum computations, we consider approximations of orthogonal matrices U by ``non-commutative convex combinations'' A of permutation matrices of the…

It is well-known, and was first established by Knuth in 1969, that the number of 321-avoiding permutations is equal to that of 132-avoiding permutations. In the literature one can find many subsequent bijective proofs of this fact. It turns…

Let G be a finite group, K a normal subgroup of G and H a subgroup such that G = HK, and set L = H \cap K. Suppose \theta \in Irr K and \phi \in Irr L, and \phi\ occurs in \theta_L with multiplicity n > 0. A projective representation of…

We study a birational map associated to any finite poset P. This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set of order ideals of P. Classical rowmotion…

We prove that the grades of simple modules indexed by boolean permutations, over the incidence algebra of the symmetric group with respect to the Bruhat order, are given by Lusztig's a-function. Our arguments are combinatorial, and include…

We present a simple bijection between permutation matrices and descending plane partitions without special parts. This bijection is already mentioned in work of P. Lalonde (without giving the details); it involves the inversion words of…

Let A be a local commutative principal ideal ring. We study the double coset space of GL(n,A) with respect to the subgroup of upper triangular matrices. Geometrically, these cosets describe the relative position of two full flags of free…

The superspace ring $\Omega_n$ is a rank $n$ polynomial ring tensor a rank $n$ exterior algebra. Using an extension of the Vandermonde determinant to $\Omega_n$, the authors previously defined a family of doubly graded quotients…

A hyperbinary partition of the nonnegative integer n is a partition where every part is a power of 2 and every part appears at most twice. We give three applications of the length generating function for such partitions, denoted by h_q(n).…

In this paper, we extend the study of the quantum $k$-Bruhat order initiated in the work of Benedetti, Bergeron, Colmenarejo, Saliola, and Sottile concerning the quantum Murnaghan-Nakayama rule. Specifically, identifying maximal chains in…

In this note a bijection is constructed between the set of partitions of n simultaneously s-regular and t-distinct, and those simultaneously t-regular and s-distinct. Some implications of the map are discussed. As a generalized version of…