Showing papers on "Symmetric group published in 2021"

Hadamard-Free Circuits Expose the Structure of the Clifford Group

Abstract: The Clifford group plays a central role in quantum randomized benchmarking, quantum tomography, and error correction protocols. Here we study the structural properties of this group. We show that any Clifford operator can be uniquely written in the canonical form $F_{1}HSF_{2}$ , where $H$ is a layer of Hadamard gates, $S$ is a permutation of qubits, and $F_{i}$ are parameterized Hadamard-free circuits chosen from suitable subgroups of the Clifford group. Our canonical form provides a one-to-one correspondence between Clifford operators and layered quantum circuits. We report a polynomial-time algorithm for computing the canonical form. We employ this canonical form to generate a random uniformly distributed $n$ -qubit Clifford operator in runtime $O(n^{2})$ . The number of random bits consumed by the algorithm matches the information-theoretic lower bound. A surprising connection is highlighted between random uniform Clifford operators and the Mallows distribution on the symmetric group. The variants of the canonical form, one with a short Hadamard-free part and one allowing a circuit depth $9n$ implementation of arbitrary Clifford unitaries in the Linear Nearest Neighbor architecture are also discussed. Finally, we study computational quantum advantage where a classical reversible linear circuit can be implemented more efficiently using Clifford gates, and show an explicit example where such an advantage takes place.

Surface words are determined by word measures on groups

Abstract: Every word w in a free group naturally induces a probability measure on every compact group G. For example, if w = [x, y] is the commutator word, a random element sampled by the w-measure is given by the commutator [g, h] of two independent, Haar-random elements of G. Back in 1896, Frobenius showed that if G is a finite group and ψ an irreducible character, then the expected value of ψ([g, h]) is $${1 \over {\psi \left(e \right)}}$$ . This is true for any compact group, and completely determines the [x, y]-measure on these groups. An analogous result holds with the commutator word replaced by any surface word. We prove a converse to this theorem: if w induces the same measure as [x, y] on every compact group, then, up to an automorphism of the free group, w is equal to [x, y]. The same holds when [x, y] is replaced by any surface word. The proof relies on the analysis of word measures on unitary groups and on orthogonal groups, which appears in separate papers, and on new analysis of word measures on generalized symmetric groups that we develop here.

Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group

Abstract: Ordering permutations by containment of inversion sets yields a fascinating partial order on the symmetric group: the weak order. This partial order is, among other things, a semidistributive lattice. As a consequence, every permutation has a canonical representation as a join of other permutations. Combinatorially, these canonical join representations can be modeled in terms of arc diagrams. Moreover, these arc diagrams also serve as a model to understand quotient lattices of the weak order. A particularly well-behaved quotient lattice of the weak order is the well-known Tamari lattice, which appears in many seemingly unrelated areas of mathematics. The arc diagrams representing the members of the Tamari lattices are better known as noncrossing partitions. Recently, the Tamari lattices were generalized to parabolic quotients of the symmetric group. In this article, we undertake a structural investigation of these parabolic Tamari lattices, and explain how modified arc diagrams aid the understanding of these lattices.

Equivariant Hilbert series for hierarchical Models

Abstract: Toric ideals to hierarchical models are invariant under the action of a product of symmetric groups. Taking the number of factors, say m, into account, we introduce and study invariant filtrations and their equivariant Hilbert series. We present a condition that guarantees that the equivariant Hilbert series is a rational function in m+1 variables with rational coefficients. Furthermore we give explicit formulas for the rational functions with coefficients in a number field and an algorithm for determining the rational functions with rational coefficients. A key is to construct finite automata that recognize languages corresponding to invariant filtrations.

The Spread of Almost Simple Classical Groups

Abstract: Every finite simple group can be generated by two elements, and in 2000, Guralnick and Kantor resolved a 1962 question of Steinberg by proving that in a finite simple group every nontrivial element belongs to a generating pair. Groups with this property are said to be $\frac{3}{2}$-generated. Which finite groups are $\frac{3}{2}$-generated? Every proper quotient of a $\frac{3}{2}$-generated group is cyclic, and in 2008, Breuer, Guralnick and Kantor made the striking conjecture that this condition alone provides a complete characterisation of the finite groups with this property. This conjecture has recently been reduced to the almost simple groups and results of Piccard (1939) and Woldar (1994) show that the conjecture is true for almost simple groups whose socles are alternating or sporadic groups. Therefore, the central focus is now on the almost simple groups of Lie type. In this monograph we prove a strong version of this conjecture for almost simple classical groups, building on earlier work of Burness and Guest (2013) and the author (2017). More precisely, we show that every relevant almost simple classical group has uniform spread at least two, unless it is isomorphic to the symmetric group of degree six. We also prove that the uniform spread of these groups tends to infinity if the size of the underlying field tends to infinity. To prove these results, we are guided by a probabilistic approach introduced by Guralnick and Kantor. This requires a detailed analysis of automorphisms, fixed point ratios and subgroup structure of almost simple classical groups, so the first half of this monograph is dedicated to these general topics. In particular, we give a general exposition of the useful technique of Shintani descent, which plays an important role throughout.

Positive maps and trace polynomials from the symmetric group

Abstract: With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions that involve matrix monomials Xα1,…,Xαr and their traces tr(Xα1,…,Xαr). Our method rests on translating the action of the symmetric group on tensor product spaces into that of matrix multiplication. As a result, we extend the polarized Cayley–Hamilton identity to an operator inequality on the positive cone, characterize the set of multilinear equivariant positive maps in terms of Werner state witnesses, and construct permutation polynomials and tensor polynomial identities on tensor product spaces. We give connections to concepts in quantum information theory and invariant theory.

Boolean product polynomials, Schur positivity, and Chern plethysm

Abstract: Let $1\leq k \leq n$ and let $X_n = (x_1, \dots, x_n)$ be a list of $n$ variables. The {\em Boolean product polynomial} $B_{n,k}(X_n)$ is the product of the linear forms $\sum_{i \in S} x_i$ where $S$ ranges over all $k$-element subsets of $\{1, 2, \dots, n\}$. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call {\em Chern plethysm}. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a combinatorial or representation theoretic proof. We relate the polynomials $B_{n,k}(X_n)$ for certain $k$ to other combinatorial objects including derangements, positroids, alternating sign matrices, and reverse flagged fillings of a partition shape. We also relate $B_{n,n-1}(X_n)$ to a bigraded action of the symmetric group $\mathfrak{S}_n$ on a divergence free quotient of superspace.

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

Abstract: We give a construction of general holomorphic quarter BPS operators in $$ \mathcal{N} $$ = 4 SYM at weak coupling with U(N) gauge group at finite N. The construction employs the Mobius 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.

Pinnacle Set Properties

Abstract: Let pi = pi_1 pi_2 ... pi_n be a permutation in the symmetric group S_n written in one-line notation. The pinnacle set of pi, denoted Pin pi, is the set of all pi_i such that pi_{i-1} pi_{i+1}. This is an analogue of the well-studied peak set of pi where one considers values rather than positions. The pinnacle set was introduced by Davis, Nelson, Petersen, and Tenner who showed that it has many interesting properties. In particular, they proved that the number of subsets of [n] = {1, 2, ..., n} which can be the pinnacle set of some permutation is a binomial coefficient. Their proof involved a bijection with lattice paths and was somewhat involved. We give a simpler demonstration of this result which does not need lattice paths. Moreover, we show that our map and theirs are different descriptions of the same function. Davis et al. also studied the number of pinnacle sets with maximum m and cardinality d which they denoted by p(m,d). We show that these integers are ballot numbers and give two proofs of this fact: one using finite differences and one bijective. Diaz-Lopez, Harris, Huang, Insko, and Nilsen found a summation formula for calculating the number of permutations in S_n having a given pinnacle set. We derive a new expression for this number which is faster to calculate in many cases. We also show how this method can be adapted to find the number of orderings of a pinnacle set which can be realized by some pi in S_n.

Combinatorial generation via permutation languages. II. Lattice congruences

Abstract: This paper deals with lattice congruences of the weak order on the symmetric group, and initiates the investigation of the cover graphs of the corresponding lattice quotients. These graphs also arise as the skeleta of the so-called quotientopes, a family of polytopes recently introduced by Pilaud and Santos [Bull. Lond. Math. Soc., 51:406–420, 2019], which generalize permutahedra, associahedra, hypercubes and several other polytopes. We prove that all of these graphs have a Hamilton path, which can be computed by a simple greedy algorithm. This is an application of our framework for exhaustively generating various classes of combinatorial objects by encoding them as permutations. We also characterize which of these graphs are vertex-transitive or regular via their arc diagrams, give corresponding precise and asymptotic counting results, and we determine their minimum and maximum degrees.

Cohomological Invariants in Positive Characteristic

Abstract: We determine the mod p cohomological invariants for several affine group schemes G in chararacteristic p. These are invariants of G-torsors with values in etale motivic cohomology, or equivalently in Kato's version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod p etale motivic cohomology of fields, extending Vial's computations of the operations on the mod p Milnor K-theory of fields.

The center of the wreath product of symmetric group algebras

Abstract: We consider the wreath product of two symmetric groups as a group of blocks permutations and we study its conjugacy classes. We give a polynomiality property for the structure coefficients of the center of the wreath product of symmetric group algebras. This allows us to recover an old result of Farahat and Higman about the polynomiality of the structure coefficients of the center of the symmetric group algebra and to generalize our recent result about the polynomiality property of the structure coefficients of the center of the hyperoctahedral group algebra.

Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence

Abstract: In this note, we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.

Whitney Numbers for Poset Cones

Abstract: Hyperplane arrangements dissect $\mathbb {R}^{n}$ into connected components called chambers, and a well-known theorem of Zaslavsky counts chambers as a sum of nonnegative integers called Whitney numbers of the first kind. His theorem generalizes to count chambers within any cone defined as the intersection of a collection of halfspaces from the arrangement, leading to a notion of Whitney numbers for each cone. This paper focuses on cones within the braid arrangement, consisting of the reflecting hyperplanes xi = xj inside $\mathbb {R}^{n}$ for the symmetric group, thought of as the type An− 1 reflection group. Here, We interpret this refinement for all posets as counting linear extensions according to a statistic that generalizes the number of left-to-right maxima of a permutation. When the poset is a disjoint union of chains, we interpret this refinement differently, using Foata’s theory of cycle decomposition for multiset permutations, leading to a simple generating function compiling these Whitney numbers.

Completely Reachable Automata, Primitive Groups and the State Complexity of the Set of Synchronizing Words

Abstract: We give a new characterization of primitive permutation groups tied to the notion of completely reachable automata. Also, we introduce sync-maximal permutation groups tied to the state complexity of the set of synchronizing words of certain associated automata and show that they are contained between the 2-homogeneous and the primitive groups. Lastly, we define k-reachable groups in analogy with synchronizing groups and motivated by our characterization of primitive permutation groups. But the results show that a k-reachable permutation group of degree n with \(6 \le k \le n - 6\) is either the alternating or the symmetric group.

Baxterisation of the fused Hecke algebra and R-matrices with gl(N)-symmetry

Abstract: We give an explicit Baxterisation formula for the fused Hecke algebra and its classical limit for the algebra of fused permutations. These algebras replace the Hecke algebra and the symmetric group in the Schur–Weyl duality theorems for the symmetrised powers of the fundamental representation of gl(N) and their quantum version. So the Baxterisation formulas presented in this paper are applicable to the R-matrices associated with these representations. In particular, all “higher spins” representations of (classical and quantum) sl(2) are covered.

Sylow branching coefficients for symmetric groups

Abstract: Let $p\ge 5$ be a prime and let $n$ be a natural number. In this article we describe the irreducible constituents of the induced characters $\phi\big\uparrow^{\mathfrak{S}_n}$ for arbitrary linear characters $\phi$ of a Sylow $p$-subgroup of the symmetric group $\mathfrak{S}_n$, generalising earlier results of the authors. By doing so, we introduce Sylow branching coefficients for symmetric groups.

Differential operators and the symmetric groups

Abstract: In this paper, we study the action of the rational quantum Calogero-Moser system on polynomials. In this vein, we study polynomials ring over the complex field ℂ as a module over a ring of differential operators by elaborating its irreducible submodules. we endowed the polynomial ring ℂ[x 1 , …, x n ] with a differential structure by using directly the action of the Weyl algebra associated with the ring of symmetric polynomial ℂ[x 1 , …, x n ] Sn after a localization. Then we study the polynomials representation of the ring of invariant differential operators under the symmetric group. We use the representation theory of symmetric groups to exhibit the generators of its simple components.

Rigid commutators and a normalizer chain

Abstract: The notion of rigid commutators is introduced to determine the sequence of the logarithms of the indices of a certain normalizer chain in the Sylow 2-subgroup of the symmetric group on $$2^n$$ letters. The terms of this sequence are proved to be those of the partial sums of the partitions of an integer into at least two distinct parts, that relates to a famous Euler’s partition theorem.

Singularities of Schubert Varieties within a Right Cell

Abstract: We describe an algorithm which pattern embeds, in the sense of Woo-Yong, any Bruhat interval of a symmetric group into an interval whose extremes lie in the same right Kazhdan-Lusztig cell. This apparently harmless fact has applications in finding examples of reducible associated varieties of $\mathfrak{sl}_n$-highest weight modules, as well as in the study of $W$-graphs for symmetric groups, and in comparing various bases of irreducible representations of the symmetric group or its Hecke algebra. For example, we are able to systematically produce many negative answers to a question from the 1980s of Borho-Brylinski and Joseph, which had been settled by Williamson via computer calculations only in 2014.

Automorphism groups of some families of bipartite graphs

Abstract: This paper discusses the automorphism group of a class of weakly semiregular bipartite graphs and its subclass called WSB END graphs. It also tries to analyse the automorphism group of the SM sum graphs and SM balancing graphs. These graphs are weakly semiregular bipartite graphs too. The SM sum graphs are particular cases of bipartite Kneser graphs. The bipartite Kneser type graphs are defined on n -sets for a fixed positive integer n . The automorphism groups of the bipartite Kneser type graphs are related to that of weakly semiregular bipartite graphs. Weakly semiregular bipartite graphs in which the neighbourhoods of the vertices in the SD part having the same degree sequence, possess non trivial automorphisms. The automorphism groups of SM sum graphs are isomorphic to the symmetric groups. The relationship between the automorphism groups of SM balancing graphs and symmetric groups are established here. It has been observed by using the well known algorithm Nauty, that the size of automorphism groups of SM balancing graphs are prodigious. Every weakly semiregular bipartite graphs with k-NSD subparts has a matching which saturates the smaller partition.

Symmetry reduction in AM/GM-based optimization

Abstract: The arithmetic mean/geometric mean-inequality (AM/GM-inequality) facilitates classes of non-negativity certificates and of relaxation techniques for polynomials and, more generally, for exponential sums. Here, we present a first systematic study of the AM/GM-based techniques in the presence of symmetries under the linear action of a finite group. We prove a symmetry-adapted representation theorem and develop techniques to reduce the size of the resulting relative entropy programs. We study in more detail the complexity gain in the case of the symmetric group. In this setup, we can show in particular certain stabilization results. We exhibit several sequences of examples in growing dimensions where the size of the problem stabilizes. Finally, we provide some numerical results, emphasizing the computational speed-up.