37,182 research outputs found

### Entanglement in a second order topological insulator on a square lattice

In a $d$-dimensional topological insulator of order $d$, there are zero
energy states on its corners which have close relationship with its
entanglement behaviors. We studied the bipartite entanglement spectra for
different subsystem shapes and found that only when the entanglement boundary
has corners matching the lattice, exact zero modes exist in the entanglement
spectrum corresponding to the zero energy states caused by the same physical
corners. We then considered finite size systems in which case these corner
states are coupled together by long range hybridizations to form a multipartite
entangled state. We proposed a scheme to calculate the quadripartite
entanglement entropy on the square lattice, which is well described by a
four-sites toy model and thus provides another way to identify the higher order
topological insulators from the multipartite entanglement point of view.Comment: 5 pages, 3 figure

### Cyclotomy and permutation polynomials of large indices

We use cyclotomy to design new classes of permutation polynomials over finite
fields. This allows us to generate many classes of permutation polynomials in
an algorithmic way. Many of them are permutation polynomials of large indices

### Promotion operator on rigged configurations of type A

Recently, the analogue of the promotion operator on crystals of type A under
a generalization of the bijection of Kerov, Kirillov and Reshetikhin between
crystals (or Littlewood--Richardson tableaux) and rigged configurations was
proposed. In this paper, we give a proof of this conjecture. This shows in
particular that the bijection between tensor products of type A_n^{(1)}
crystals and (unrestricted) rigged configurations is an affine crystal
isomorphism.Comment: 37 page

### Promotion and evacuation on standard Young tableaux of rectangle and staircase shape

(Dual-)promotion and (dual-)evacuation are bijections on SYT(\lambda) for any
partition \lambda. Let c^r denote the rectangular partition (c,...,c) of height
r, and let sc_k (k > 2) denote the staircase partition (k,k-1,...,1). B.
Rhoades showed representation-theoretically that promotion on SYT(c^r) exhibits
the cyclic sieving phenomenon (CSP). In this paper, we demonstrate a promotion-
and evacuation-preserving embedding of SYT(sc_k) into SYT(k^{k+1}). This arose
from an attempt to demonstrate the CSP of promotion action on SYT(sc_k).Comment: 14 pages, typos correcte

### On the number of $N$-free elements with prescribed trace

In this paper we derive a formula for the number of $N$-free elements over a
finite field $\mathbb{F}_q$ with prescribed trace, in particular trace zero, in
terms of Gaussian periods. As a consequence, we derive a simple explicit
formula for the number of primitive elements, in quartic extensions of Mersenne
prime fields, having absolute trace zero. We also give a simple formula in the
case when $Q = (q^m-1)/(q-1)$ is prime. More generally, for a positive integer
$N$ whose prime factors divide $Q$ and satisfy the so called semi-primitive
condition, we give an explicit formula for the number of $N$-free elements with
arbitrary trace. In addition we show that if all the prime factors of $q-1$
divide $m$, then the number of primitive elements in $\mathbb{F}_{q^m}$, with
prescribed non-zero trace, is uniformly distributed. Finally we explore the
related number, $P_{q, m, N}(c)$, of elements in $\mathbb{F}_{q^m}$ with
multiplicative order $N$ and having trace $c \in \mathbb{F}_q$. Let $N \mid
q^m-1$ such that $L_Q \mid N$, where $L_Q$ is the largest factor of $q^m-1$
with the same radical as that of $Q$. We show there exists an element in
$\mathbb{F}_{q^m}^*$ of (large) order $N$ with trace $0$ if and only if $m \neq
2$ and $(q,m) \neq (4,3)$. Moreover we derive an explicit formula for the
number of elements in $\mathbb{F}_{p^4}$ with the corresponding large order
$L_Q = 2(p+1)(p^2+1)$ and having absolute trace zero, where $p$ is a Mersenne
prime

### A probabilistic approach to value sets of polynomials over finite fields

In this paper we study the distribution of the size of the value set for a
random polynomial with degree at most $q-1$ over a finite field $\mathbb{F}_q$.
We obtain the exact probability distribution and show that the number of
missing values tends to a normal distribution as $q$ goes to infinity. We
obtain these results through a study of a random $r$-th order cyclotomic
mappings. A variation on the size of the union of some random sets is also
considered

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