Showing papers on "Symmetric group published in 2018"

Permutation Groups and Cartesian Decompositions

Abstract: Permutation groups, their fundamental theory and applications are discussed in this introductory book. It focuses on those groups that are most useful for studying symmetric structures such as graphs, codes and designs. Modern treatments of the O'Nan–Scott theory are presented not only for primitive permutation groups but also for the larger families of quasiprimitive and innately transitive groups, including several classes of infinite permutation groups. Their precision is sharpened by the introduction of a cartesian decomposition concept. This facilitates reduction arguments for primitive groups analogous to those, using orbits and partitions, that reduce problems about general permutation groups to primitive groups. The results are particularly powerful for finite groups, where the finite simple group classification is invoked. Applications are given in algebra and combinatorics to group actions that preserve cartesian product structures. Students and researchers with an interest in mathematical symmetry will find the book enjoyable and useful.

Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations

Abstract: Reed-Muller (RM) and polar codes are a class of capacity-achieving channel coding schemes with the same factor graph representation. Low-complexity decoding algorithms fall short in providing a good error-correction performance for RM and polar codes. Using the symmetric group of RM and polar codes, the specific decoding algorithm can be carried out on multiple permutations of the factor graph to boost the error-correction performance. However, this approach results in high decoding complexity. In this paper, we first derive the total number of factor graph permutations on which the decoding can be performed. We further propose a successive permutation (SP) scheme which finds the permutations on the fly, thus the decoding always progresses on a single factor graph permutation. We show that SP can be used to improve the error-correction performance of RM and polar codes under successive-cancellation (SC) and SC list (SCL) decoding, while keeping the memory requirements of the decoders unaltered. Our results for RM and polar codes of length 128 and rate 0.5 show that when SP is used and at a target frame error rate of 10−4, up to 0.5 dB and 0.1 dB improvement can be achieved for RM and polar codes respectively.

Large N limit of irreducible tensor models: O(N) rank-3 tensors with mixed permutation symmetry

Abstract: It has recently been proven that in rank three tensor models, the antisymmetric and symmetric traceless sectors both support a large N expansion dominated by melon diagrams [1]. We show how to extend these results to the last irreducible O(N) tensor representation available in this context, which carries a two-dimensional representation of the symmetric group S3. Along the way, we emphasize the role of the irreducibility condition: it prevents the generation of vector modes which are not compatible with the large N scaling of the tensor interaction. This example supports the conjecture that a melonic large N limit should exist more generally for higher rank tensor models, provided that they are appropriately restricted to an irreducible subspace.

On semisimplification of tensor categories

Abstract: We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic $p$ in terms of representations of the normnalizer of its Sylow $p$-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group $S_{n+p}$ in characteristic $p$, where $0\le n\le p-1$, and of the Deligne category $\underline{\rm Rep}^{\rm ab}S_t$, where $t\in \Bbb N$. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of $\mathfrak{sl}_2$. Finally, we study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). In the appendix, we classify categorifications of the Grothendieck ring of representations of $SO(3)$ and its truncations.

Fixing Numbers of Graphs and Groups

Abstract: The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $S$ such that only the trivial automorphism of $G$ fixes every vertex in $S$. The fixing set of a group $\Gamma$ is the set of all fixing numbers of finite graphs with automorphism group $\Gamma$. Several authors have studied the distinguishing number of a graph, the smallest number of labels needed to label $G$ so that the automorphism group of the labeled graph is trivial. The fixing number can be thought of as a variation of the distinguishing number in which every label may be used only once, and not every vertex need be labeled. We characterize the fixing sets of finite abelian groups, and investigate the fixing sets of symmetric groups.

Boolean degree 1 functions on some classical association schemes

Abstract: We investigate Boolean degree 1 functions for several classical association schemes, including Johnson graphs, Grassmann graphs, graphs from polar spaces, and bilinear forms graphs, as well as some other domains such as multislices (Young subgroups of the symmetric group). In some settings, Boolean degree 1 functions are also known as \textit{completely regular strength 0 codes of covering radius 1}, \textit{Cameron--Liebler line classes}, and \textit{tight sets}. We classify all Boolean degree $1$ functions on the multislice. On the Grassmann scheme $J_q(n, k)$ we show that all Boolean degree $1$ functions are trivial for $n \geq 5$, $k, n-k \geq 2$ and $q \in \{ 2, 3, 4, 5 \}$, and that for general $q$, the problem can be reduced to classifying all Boolean degree $1$ functions on $J_q(n, 2)$. We also consider polar spaces and the bilinear forms graphs, giving evidence that all Boolean degree $1$ functions are trivial for appropriate choices of the parameters.

Arithmetic functions in short intervals and the symmetric group

Abstract: We consider the variance of sums of arithmetic functions over random short intervals in the function field setting. Based on the analogy between factorizations of random elements of $\mathbb{F}_q[T]$ into primes and the factorizations of random permutations into cycles, we give a simple but general formula for these variances in the large $q$ limit for arithmetic functions that depend only upon factorization structure. From this we derive new estimates, quickly recover some that are already known, and make new conjectures in the setting of the integers. In particular we make the combinatorial observation that any function of this sort can be decomposed into a sum of functions $u$ and $v$, depending on the size of the short interval, with $u$ making a negligible contribution to the variance, and $v$ asymptotically contributing diagonal terms only. This variance evaluation is closely related to the appearance of random matrix statistics in the zeros of families of L-functions and sheds light on the arithmetic meaning of this phenomenon.

Numerical Computation of Galois Groups

Abstract: The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. We give numerical methods to compute the Galois group and study it when it is not the full symmetric group. One algorithm computes generators, while the other studies its structure as a permutation group. We illustrate these algorithms with examples using a Macaulay2 package we are developing that relies upon Bertini to perform monodromy computations.

Reduction of triangulated categories and Maximal Modification Algebras for cA_n singularities

Abstract: In this paper we define and study triangulated categories in which the Homspaces have Krull dimension at most one over some base ring (hence they have a natural 2-step filtration), and each factor of the filtration satisfies some Calabi–Yau type property. If C is such a category, we say that C is Calabi–Yau with dim C ≤ 1. We extend the notion of Calabi–Yau reduction to this setting, and prove general results which are an analogue of known results in cluster theory. Such categories appear naturally in the setting of Gorenstein singularities in dimension three as the stable categories CM R of Cohen–Macaulay modules. We explain the connection between Calabi–Yau reduction of CM R and both partial crepant resolutions and Q-factorial terminalizations of Spec R, and we show under quite general assumptions that Calabi–Yau reductions exist. In the remainder of the paper we focus on complete local cAn singularities R. By using a purely algebraic argument based on Calabi–Yau reduction of CM R, we give a complete classification of maximal modifying modules in terms of the symmetric group, generalizing and strengthening results in [7, 10], where we do not need any restriction on the ground field. We also describe the mutation of modifying modules at an arbitrary (not necessarily indecomposable) direct summand. As a corollary when k D C we obtain many autoequivalences of the derived category of the Q-factorial terminalizations of Spec R.

Billiards and Tilting Characters for SL3

Abstract: We formulate a conjecture for the second generation characters of indecomposable tilting modules for ${\rm SL}_3$. This gives many new conjectural decomposition numbers for symmetric groups. Our conjecture can be interpreted as saying that these characters are governed by a discrete dynamical system ("billiards bouncing in alcoves"). The conjecture implies that decomposition numbers for symmetric groups display (at least) exponential growth.

Large $N$ limit of irreducible tensor models: $O(N)$ rank-$3$ tensors with mixed permutation symmetry

Abstract: It has recently been proven that in rank three tensor models, the anti-symmetric and symmetric traceless sectors both support a large $N$ expansion dominated by melon diagrams [arXiv:1712.00249 [hep-th]]. We show how to extend these results to the last irreducible $O(N)$ tensor representation available in this context, which carries a two-dimensional representation of the symmetric group $S_3$. Along the way, we emphasize the role of the irreducibility condition: it prevents the generation of vector modes which are not compatible with the large $N$ scaling of the tensor interaction. This example supports the conjecture that a melonic large $N$ limit should exist more generally for higher rank tensor models, provided that they are appropriately restricted to an irreducible subspace.

Blocks of symmetric groups, semicuspidal KLR algebras and zigzag Schur-Weyl duality

Abstract: We prove Turner's conjecture, which describes the blocks of the Hecke algebras of the symmetric groups up to derived equivalence as certain explicit Turner double algebras. Turner doubles are Schur-algebra-like `local' objects, which replace wreath products of Brauer tree algebras in the context of the Brou\'e abelian defect group conjecture for blocks of symmetric groups with non-abelian defect groups. The main tools used in the proof are generalized Schur algebras corresponding to wreath products of zigzag algebras and imaginary semicuspidal quotients of affine KLR algebras.

Galois theory for general systems of polynomial equations

Abstract: We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel--Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive, similarly to what Sottile and White conjectured in Schubert calculus. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.

Springer correspondence for the split symmetric pair in type $A$

Abstract: In this paper we establish Springer correspondence for the symmetric pair using Fourier transform, parabolic induction functor, and a nearby cycle sheaf construction. As an application of our results we see that the cohomology of Hessenberg varieties can be expressed in terms of irreducible representations of Hecke algebras of symmetric groups at . Conversely, we see that the irreducible representations of Hecke algebras of symmetric groups at arise in geometry.

Chromatic symmetric functions via the group algebra of $S_n$

Abstract: We prove some Schur positivity results for the chromatic symmetric function $X_G$ of a (hyper)graph $G$, using connections to the group algebra of the symmetric group. The first such connection works for (hyper)forests $F$: we describe the Schur coefficients of $X_F$ in terms of eigenvalues of a product of Hermitian idempotents in the group algebra, one factor for each edge (a more general formula of similar shape holds for all chordal graphs). Our main application of this technique is to prove a conjecture of Taylor on the Schur positivity of certain $X_F$, which implies Schur positivity of the formal group laws associated to various combinatorial generating functions. We also introduce the pointed chromatic symmetric function $X_{G,v}$ associated to a rooted graph $(G,v)$. We prove that if $X_{G,v}$ and $X_{H,w}$ are positive in the generalized Schur basis of Strahov, then the chromatic symmetric function of the wedge sum of $(G,v)$ and $(H,w)$ is Schur positive.

Iterates of generic polynomials and generic rational functions

Abstract: In 1985, Odoni showed that in characteristic $0$ the Galois group of the $n$-th iterate of the generic polynomial with degree $d$ is as large as possible. That is, he showed that this Galois group is the $n$-th wreath power of the symmetric group $S_d$. We generalize this result to positive characteristic, as well as to the generic rational function. These results can be applied to prove certain density results in number theory, two of which are presented here. This work was partially completed by the late R.W.K. Odoni in an unpublished paper.

Infinite families of monogenic trinomials and their Galois groups

Abstract: Let n ∈ ℤ with n ≥ 3. Let Sn and An denote, respectively, the symmetric group and alternating group on n letters. Let m be an indeterminate, and define fm(x) := xn + a(m,n)x + b(m,n), where a(m,n),b(m,n) are certain prescribed forms in m. For a certain set of these forms, we show unconditionally that there exist infinitely many primes p such that fp(x) is irreducible over ℚ, Galℚ(fp) = Sn, and the fields K = ℚ(𝜃) are distinct and monogenic, where fp(𝜃) = 0. Using a different set of forms, we establish a similar result for all square-free values of n ≡ 1(mod 4), with 5 ≤ n ≤ 401, and any positive integer value of m for which a(m,n) is square-free. Additionally, in this case, we prove that Galℚ(fp) = An. Finally, we show that these results can be extended under the assumption of the abc-conjecture. Our methods make use of recent results of Helfgott and Pasten.

On the existence of tableaux with given modular major index

Abstract: We provide simple necessary and sufficient conditions for the existence of a standard Young tableau of a given shape and major index $r$ mod $n$, for all $r$. Our result generalizes the $r=1$ case due essentially to (1974) and proves a recent conjecture due to Sundaram (2016) for the $r=0$ case. A byproduct of the proof is an asymptotic equidistribution result for "almost all" shapes. The proof uses a representation-theoretic formula involving Ramanujan sums and normalized symmetric group character estimates. Further estimates involving "opposite" hook lengths are given which are well-adapted to classifying which partitions $\lambda \vdash n$ have $f^\lambda \leq n^d$ for fixed $d$. We also give a new proof of a generalization of the hook length formula due to Fomin-Lulov (1995) for symmetric group characters at rectangles. We conclude with some remarks on unimodality of symmetric group characters.

Thompson groups for systems of groups, and their finiteness properties

Abstract: We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pure braid groups. We give new examples of families of groups that admit a cloning system and study how the finiteness properties of the resulting generalized Thompson group depend on those of the original groups. The main new examples here include upper triangular matrix groups, mock reflection groups, and loop braid groups. For generalized Thompson groups of upper triangular matrix groups over rings of S-integers of global function fields, we develop new methods for (dis-)proving finiteness properties, and show that the finiteness length of the generalized Thompson group is exactly the limit inferior of the finiteness lengths of the groups in the family.

Structural aspects of twin and pure twin groups

Abstract: The twin group $T_n$ is a Coxeter group generated by $n-1$ involutions and the pure twin group $PT_n$ is the kernel of the natural surjection of $T_n$ onto the symmetric group on $n$ letters. In this paper, we investigate structural aspects of twin and pure twin groups. We prove that the twin group $T_n$ decomposes into a free product with amalgamation for $n>4$. It is shown that the pure twin group $PT_n$ is free for $n=3,4$, and not free for $n\ge 6$. We determine a generating set for $PT_n$, and give an upper bound for its rank. We also construct a natural faithful representation of $T_4$ into $\operatorname{Aut}(F_7)$. In the end, we propose virtual and welded analogues of these groups and some directions for future work.

A Novel Technique for the Construction of Safe Substitution Boxes Based on Cyclic and Symmetric Groups

Abstract: In the literature, different algebraic techniques have been applied on Galois field to construct substitution boxes. In this paper, instead of Galois field , we use a cyclic group in the formation of proposed substitution box. The construction proposed S-box involves three simple steps. In the first step, we introduce a special type of transformation of order 255 to generate . Next, we adjoin to and write the elements of in matrix to destroy the initial sequence . In the step, the randomness in the data is increased by applying certain permutations of the symmetric group on rows and columns of the matrix. In the last step we consider the symmetric group , and positions of the elements of the matrix obtained in step 2 are changed by its certain permutations to construct the suggested S-box. The strength of our S-box to work against cryptanalysis is checked through various tests. The results are then compared with the famous S-boxes. The comparison shows that the ability of our S-box to create confusion is better than most of the famous S-boxes.

Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II

Abstract: We present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to the construction of Fast Fourier Transform algorithms we make explicit use of the path algebra connection to the construction of Gel’fand–Tsetlin bases and work in the setting of quivers. We relate this framework to the construction of a configuration space derived from a Bratteli diagram. In this setting the complexity of an algorithm for computing a Fourier transform reduces to the calculation of the dimension of the associated configuration space. Our methods give improved upper bounds for computing the Fourier transform for the general linear groups over finite fields, the classical Weyl groups, and homogeneous spaces of finite groups, while also recovering the best known algorithms for the symmetric group.

Minimax Rates and Efficient Algorithms for Noisy Sorting

Abstract: There has been a recent surge of interest in studying permutation-based models for ranking from pairwise comparison data. Despite being structurally richer and more robust than parametric ranking models, permutation-based models are less well understood statistically and generally lack efficient learning algorithms. In this work, we study a prototype of permutation-based ranking models, namely, the noisy sorting model. We establish the optimal rates of learning the model under two sampling procedures. Furthermore, we provide a fast algorithm to achieve near-optimal rates if the observations are sampled independently. Along the way, we discover properties of the symmetric group which are of theoretical interest.