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Journal articles on the topic 'Group theory and generalizations – Permutation groups – General theory for finite groups'
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1
Tornier, Stephan. "Prime localizations of Burger–Mozes-type groups." Journal of Group Theory 21, no. 2 (2018): 229–40. http://dx.doi.org/10.1515/jgth-2017-0036.
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AbstractThis article concerns Burger–Mozes universal groups acting on regular trees locally like a given permutation group of finite degree. We also consider locally isomorphic generalizations of the former due to Le Boudec and Lederle. For a large class of such permutation groups and primespwe determine their localp-Sylow subgroups as well as subgroups of theirp-localization, which is identified as a group of the same type in certain cases.
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Kornyak, Vladimir. "Modeling Quantum Behavior in the Framework of Permutation Groups." EPJ Web of Conferences 173 (2018): 01007. http://dx.doi.org/10.1051/epjconf/201817301007.
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Quantum-mechanical concepts can be formulated in constructive finite terms without loss of their empirical content if we replace a general unitary group by a unitary representation of a finite group. Any linear representation of a finite group can be realized as a subrepresentation of a permutation representation. Thus, quantum-mechanical problems can be expressed in terms of permutation groups. This approach allows us to clarify the meaning of a number of physical concepts. Combining methods of computational group theory with Monte Carlo simulation we study a model based on representations of permutation groups.
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Niemenmaa, Markku. "Decomposition of Transformation Groups of Permutation Machines." Fundamenta Informaticae 10, no. 4 (1987): 363–67. http://dx.doi.org/10.3233/fi-1987-10403.
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By a permutation machine we mean a triple (Q,S,F), where Q and S are finite sets and F is a function Q × S → Q which defines a permutation on Q for every element from S. These permutations generate a permutation group G and by considering the structure of G we can obtain efficient ways to decompose the transformation group (Q,G). In this paper we first consider the situation where G is half-transitive and after this we show how to use our result in the general non-transitive case.
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Lulek, B., T. Lulek, J. Biel, and R. Chatterjee. "Racah–Wigner approach to standardization of permutation representations for finite groups." Canadian Journal of Physics 63, no. 8 (1985): 1065–73. http://dx.doi.org/10.1139/p85-174.
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The problem of equivalence of permutation representations of the finite groups is discussed in terms of transforms by bijections of the carrier sets and by group automorphisms. A formal description of a transformation between equivalent representations is given, and a standard form for an arbitrary permutation representation is proposed. The standardization is achieved through the canonical realization of transitive representations and of imprimitivity sets on the left cosets of the group with respect to an appropriate stability subgroup. The purpose of this paper is to pave the way for a systematic formulation of permutation representations, analogous to the Racah algebra of angular momentum theory, which will be useful to multicentre problems of quantum mechanics and statistical physics.
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Li, Chi-Kwong, and Wayne Whitney. "Closed Symmetric Overgroups of Sn in On." Canadian Mathematical Bulletin 39, no. 1 (1996): 83–94. http://dx.doi.org/10.4153/cmb-1996-011-7.
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AbstractA norm on ℝn is said to be permutation invariant if its value is preserved under permutation of the coordinates of a vector. The isometry group of such a norm must be closed, contain Sn and —I, and be conjugate to a subgroup of On, the orthogonal group. Motivated by this, we are interested in classifying all closed groups G such that 〈—I,Sn〉 < G < On. We use the theory of Lie groups to classify all possible infinite groups G, and use the theory of finite reflection groups to classify all possible finite groups G. In keeping with the original motivation, all groups arising are shown to be isometry groups. This completes the work of Gordon and Lewis, who studied the same problem and obtained the results for n ≥ 13.
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BAMBERG, JOHN, TOMASZ POPIEL, and CHERYL E. PRAEGER. "SIMPLE GROUPS, PRODUCT ACTIONS, AND GENERALIZED QUADRANGLES." Nagoya Mathematical Journal 234 (September 14, 2017): 87–126. http://dx.doi.org/10.1017/nmj.2017.35.
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The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized quadrangles that admit a collineation group$G$preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on$G$, the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that$G$cannot haveholomorph compoundO’Nan–Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in nonabelian finite simple groups and fixities of primitive permutation groups.
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ALP, MURAT, and CHRISTOPHER D. WENSLEY. "ENUMERATION OF CAT1-GROUPS OF LOW ORDER." International Journal of Algebra and Computation 10, no. 04 (2000): 407–24. http://dx.doi.org/10.1142/s0218196700000170.
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In this paper we describe a share package [Formula: see text] of functions for computing with finite, permutation crossed modules, cat1-groups and their morphisms, written using the [Formula: see text] group theory programming language. The category XMod of crossed modules is equivalent to the category Cat1 of cat1-groups and we include functions emulating the functors between these categories. The monoid of derivations of a crossed module [Formula: see text] , and the corresponding monoid of sections of a cat1-group [Formula: see text] , are constructed using the Whitehead multiplication. The Whitehead group of invertible derivations, together with the group of automorphisms of [Formula: see text] , are used to construct the actor crossed module of [Formula: see text] which is the automorphism object in XMod. We include a table of the 350 isomorphism classes of cat1-structures on groups of order at most 30.
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Smith, Larry. "Modular Vector Invariants of Cyclic Permutation Representations." Canadian Mathematical Bulletin 42, no. 1 (1999): 125–28. http://dx.doi.org/10.4153/cmb-1999-014-5.
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AbstractVector invariants of finite groups (see the introduction for an explanation of the terminology) have often been used to illustrate the difficulties of invariant theory in the modular case: see, e.g., [1], [2], [4], [7], [11] and [12]. It is therefore all the more surprising that the unpleasant properties of these invariants may be derived from two unexpected, and remarkable, nice properties: namely for vector permutation invariants of the cyclic group of prime order in characteristic p the image of the transfer homomorphism is a prime ideal, and the quotient algebra is a polynomial algebra on the top Chern classes of the action.
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Zaid, Nurhidayah, Nor Haniza Sarmin, and Sanhan Muhammad Salih Khasraw. "THE APPLICATIONS OF ZERO DIVISORS OF SOME FINITE RINGS OF MATRICES IN PROBABILITY AND GRAPH THEORY." Jurnal Teknologi 83, no. 1 (2020): 127–32. http://dx.doi.org/10.11113/jurnalteknologi.v83.14936.
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Let R be a finite ring. The zero divisors of R are defined as two nonzero elements of R, say x and y where xy = 0. Meanwhile, the probability that two random elements in a group commute is called the commutativity degree of the group. Some generalizations of this concept have been done on various groups, but not in rings. In this study, a variant of probability in rings which is the probability that two elements of a finite ring have product zero is determined for some ring of matrices over integers modulo n. The results are then applied into graph theory, specifically the zero divisor graph. This graph is defined as a graph where its vertices are zero divisors of R and two distinct vertices x and y are adjacent if and only if xy = 0. It is found that the zero divisor graph of R is a directed graph.
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DOOLEY, A. H., and V. YA GOLODETS. "The geometric dimension of an equivalence relation and finite extensions of countable groups." Ergodic Theory and Dynamical Systems 29, no. 6 (2009): 1789–814. http://dx.doi.org/10.1017/s014338570800093x.
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AbstractWe say that the geometric dimension of a countable group G is equal to n if any free Borel action of G on a standard Borel probability space (X,μ), induces an equivalence relation of geometric dimension n on (X,μ) in the sense of Gaboriau. Let ℬ be the set of all finitely generated amenable groups all of whose subgroups are also finitely generated, and let 𝒜 be the subset of ℬ consisting of finite groups, torsion-free groups and their finite extensions. In this paper we study finite free products K of groups in 𝒜. The geometric dimension of any such group K is one: we prove that also geom-dim(Gf(K))=1 for any finite extension Gf(K) of K, applying the results of Stallings on finite extensions of free product groups, together with the results of Gaboriau and others in orbit equivalence theory. Using results of Karrass, Pietrowski and Solitar we extend these results to finite extensions of free groups. We also give generalizations and applications of these results to groups with geometric dimension greater than one. We construct a family of finitely generated groups {Kn}n∈ℕ,n>1, such that geom-dim(Kn)=n and geom-dim(Gf(Kn))=n for any finite extension Gf(Kn) of Kn. In particular, this construction allows us to produce, for each integer n>1, a family of groups {K(s,n)}s∈ℕ of geometric dimension n, such that any finite extension of K(s,n) also has geometric dimension n, but the finite extensions Gf(K(s,n)) are non-isomorphic, if s≠s′.
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Bermejo-Vega, Juan, and Maarten Van den Nest. "Classical simulations of Abelian-group normalizer circuits with intermediate measurements." Quantum Information and Computation 14, no. 3&4 (2014): 181–216. http://dx.doi.org/10.26421/qic14.3-4-1.
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Quantum normalizer circuits were recently introduced as generalizations of Clifford circuits: a normalizer circuit over a finite Abelian group G is composed of the quantum Fourier transform (QFT) over G, together with gates which compute quadratic functions and automorphisms. In \cite{VDNest_12_QFTs} it was shown that every normalizer circuit can be simulated efficiently classically. This result provides a nontrivial example of a family of quantum circuits that cannot yield exponential speed-ups in spite of usage of the QFT, the latter being a central quantum algorithmic primitive. Here we extend the aforementioned result in several ways. Most importantly, we show that normalizer circuits supplemented with intermediate measurements can also be simulated efficiently classically, even when the computation proceeds adaptively. This yields a generalization of the Gottesman-Knill theorem (valid for n-qubit Clifford operations) to quantum circuits described by arbitrary finite Abelian groups. Moreover, our simulations are twofold: we present efficient classical algorithms to sample the measurement probability distribution of any adaptive-normalizer computation, as well as to compute the amplitudes of the state vector in every step of it. Finally we develop a generalization of the stabilizer formalism relative to arbitrary finite Abelian groups: for example we characterize how to update stabilizers under generalized Pauli measurements and provide a normal form of the amplitudes of generalized stabilizer states using quadratic functions and subgroup cosets.
12
Ivanyos, G. "On solving systems of random linear disequations." Quantum Information and Computation 8, no. 6&7 (2008): 579–94. http://dx.doi.org/10.26421/qic8.6-7-2.
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An important special case of the hidden subgroup problem is equivalent to the hidden shift problem over abelian groups. An efficient solution to the latter problem could serve as a building block of quantum hidden subgroup algorithms over solvable groups. The main idea of a promising approach to the hidden shift problem is a reduction to solving systems of certain random disequations in finite abelian groups. By a disequation we mean a constraint of the form $f(x)\neq 0$. In our case, the functions on the left hand side are generalizations of linear functions. The input is a random sample of functions according to a distribution which is up to a constant factor uniform over the "linear" functions $f$ such that $f(u)\neq 0$ for a fixed, although unknown element $u\in A$. The goal is to find $u$, or, more precisely, all the elements $u'\in A$ satisfying the same disequations as $u$. In this paper we give a classical probabilistic algorithm which solves the problem in an abelian $p$-group $A$ in time polynomial in the sample size $N$, where $N=(\log\size{A})^{O(q^2)}$, and $q$ is the exponent of $A$.
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CAMERON, PETER J., and PRISCILA A. KAZANIDIS. "CORES OF SYMMETRIC GRAPHS." Journal of the Australian Mathematical Society 85, no. 2 (2008): 145–54. http://dx.doi.org/10.1017/s1446788708000815.
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AbstractThe core of a graph Γ is the smallest graph Δ that is homomorphically equivalent to Γ (that is, there exist homomorphisms in both directions). The core of Γ is unique up to isomorphism and is an induced subgraph of Γ. We give a construction in some sense dual to the core. The hull of a graph Γ is a graph containing Γ as a spanning subgraph, admitting all the endomorphisms of Γ, and having as core a complete graph of the same order as the core of Γ. This construction is related to the notion of a synchronizing permutation group, which arises in semigroup theory; we provide some more insight by characterizing these permutation groups in terms of graphs. It is known that the core of a vertex-transitive graph is vertex-transitive. In some cases we can make stronger statements: for example, if Γ is a non-edge-transitive graph, we show that either the core of Γ is complete, or Γ is its own core. Rank-three graphs are non-edge-transitive. We examine some families of these to decide which of the two alternatives for the core actually holds. We will see that this question is very difficult, being equivalent in some cases to unsolved questions in finite geometry (for example, about spreads, ovoids and partitions into ovoids in polar spaces).
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SORKIN, RAFAEL D., and SUMATI SURYA. "AN ANALYSIS OF THE REPRESENTATIONS OF THE MAPPING CLASS GROUP OF A MULTIGEON THREE-MANIFOLD." International Journal of Modern Physics A 13, no. 21 (1998): 3749–90. http://dx.doi.org/10.1142/s0217751x98001761.
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It is known that the distinct unitary irreducible representations (UIR's) of the mapping class group G of a three-manifold ℳ give rise to distinct quantum sectors ("θ sectors") in quantum theories of gravity based on a product space–time of the form ℝ×ℳ. In this paper, we study the UIR's of G in an effort to understand the physical implications of these quantum sectors. The mapping class group of a three-manifold which is the connected sum of ℝ3 with a finite number of irreducible primes is a semidirect product group. Following Mackey's theory of induced representations, we provide an analysis of the structure of the general finite-dimensional UIR of such a group. In the picture of quantized primes as particles (topological geons), this general group-theoretic analysis enables one to draw several qualitative conclusions about the geons' behavior in different quantum sectors, without requiring an explicit knowledge of the UIR's corresponding to the individual primes. An important general result is that the classification of the UIR's of the so-called particle subgroup (equivalently, the UIR's of G in which the slide diffeomorphisms are represented trivially) is reduced to the problem of finding the UIR's of the internal diffeomorphism groups of the individual primes. Moreover, this reduction is entirely consistent with the geon picture, in which the UIR of the internal group of a prime determines the species of the corresponding quantum geon, and the remaining freedom in the overall UIR of G expresses the possibility of choosing an arbitrary statistics (Bose, Fermi or para) for the geons of each species. For UIR's which represent the slides nontrivially, we do not provide a complete classification, but we find some new types of efforts due to the slides, including quantum breaking of internal symmetry and of particle indistinguishability. In connection with the latter, a novel kind of statistics arises which is determined by representations of proper subgroups of the permutation group, rather than of the group as a whole. Finally, we observe that for a generic three-manifold there will be an infinity of inequivalent UIR's and hence an infinity of "consistent" theories, when topology change is neglected.
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Hivert, Florent, Anne Schilling, and Nicolas M. Thiéry. "The biHecke monoid of a finite Coxeter group." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AN,..., Proceedings (2010). http://dx.doi.org/10.46298/dmtcs.2851.
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arXiv : http://arxiv.org/abs/0912.2212 International audience For any finite Coxeter group $W$, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on $W$. The construction of the biHecke monoid relies on the usual combinatorial model for the $0-Hecke$ algebra $H_0(W)$, that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it admits |W| simple and projective modules. In order to construct the simple modules, we introduce for each $w∈W$ a combinatorial module $T_w$ whose support is the interval $[1,w]_R$ in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra, with the combinatorics of descents replaced by that of blocks and of the cutting poset. Pour tout groupe de Coxeter fini $W$, nous définissons deux nouveaux objets : son ordre de coupures et son monoïde de Hecke double. L'ordre de coupures, construit au moyen d'une généralisation de la notion de bloc dans les matrices de permutations, est presque un treillis sur $W$. La construction du monoïde de Hecke double s'appuie sur le modèle combinatoire usuel de la $0-algèbre$ de Hecke $H_0(W)$, pour le groupe symétrique, l'algèbre (ou le monoïde) engendré par les opérateurs de tri par bulles élémentaires. Les auteurs ont introduit précédemment l'algèbre de Hecke-groupe, construite comme l'algèbre engendrée conjointement par les opérateurs de tri et d'anti-tri, et décrit sa théorie des représentations. Dans cet article, nous considérons le monoïde engendré par ces opérateurs. Nous montrons qu'il admet $|W|$ modules simples et projectifs. Afin de construire ses modules simples, nous introduisons pour tout $w∈W$ un module combinatoire $T_w$ dont le support est l'intervalle [$1,w]_R$ pour l'ordre faible droit. Ce module détermine une algèbre dont la théorie des représentations généralise celle de l'algèbre de Hecke groupe, en remplaçant la combinatoire des descentes par celle des blocs et de l'ordre de coupures.
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Hegarty, Peter, and Anders Martinsson. "Permutations Destroying Arithmetic Progressions in Finite Cyclic Groups." Electronic Journal of Combinatorics 22, no. 4 (2015). http://dx.doi.org/10.37236/5340.
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A permutation $\pi$ of an abelian group $G$ is said to destroy arithmetic progressions (APs) if, whenever $(a, \, b, \, c)$ is a non-trivial 3-term AP in $G$, that is $c-b=b-a$ and $a, \, b, \, c$ are not all equal, then $(\pi(a), \, \pi(b), \pi(c))$ is not an AP. In a paper from 2004, the first author conjectured that such a permutation exists of $\mathbb{Z}_n$, for all $n \not\in \{2, \, 3, \, 5, \, 7\}$. Here we prove, as a special case of a more general result, that such a permutation exists for all $n \geq n_0$, for some explicitly constructed number $n_0 \approx 1.4 \times 10^{14}$. We also construct such a permutation of $\mathbb{Z}_p$ for all primes $p > 3$ such that $p \equiv 3 \; ({\hbox{mod $8$}})$.
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GILL, NICK, BIANCA LODÀ, and PABLO SPIGA. "ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP." Nagoya Mathematical Journal, July 13, 2021, 1–40. http://dx.doi.org/10.1017/nmj.2021.6.
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Abstract Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).
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Huber, Michael. "On the existence of block-transitive combinatorial designs." Discrete Mathematics & Theoretical Computer Science Vol. 12 no. 1, Combinatorics (2010). http://dx.doi.org/10.46298/dmtcs.516.
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Combinatorics International audience Block-transitive Steiner t-designs form a central part of the study of highly symmetric combinatorial configurations at the interface of several disciplines, including group theory, geometry, combinatorics, coding and information theory, and cryptography. The main result of the paper settles an important open question: There exist no non-trivial examples with t = 7 (or larger). The proof is based on the classification of the finite 3-homogeneous permutation groups, itself relying on the finite simple group classification.
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Lewis-Brown, Christopher, and Sanjaye Ramgoolam. "Quarter-BPS states, multi-symmetric functions and set partitions." Journal of High Energy Physics 2021, no. 3 (2021). http://dx.doi.org/10.1007/jhep03(2021)153.
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Abstract We give a construction of general holomorphic quarter BPS operators in $$ \mathcal{N} $$ N = 4 SYM at weak coupling with U(N) gauge group at finite N. The construction employs the Möbius 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.