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Group Decision Diagram (GDD): A Compact Representation for Permutations

2019
*
PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE
*

In some applications,

doi:10.1609/aaai.v33i01.33012986
fatcat:jwzbihv7one6jpwhqrds7tke4a
*a*subset of*a**permutation**group*must be maintained efficiently. ... In this study, we develop*a*new data structure, called*group*decision diagram (GDD), to maintain*a*set of*permutations*. ... The authors thank Shin'ichi Minato*for*providing source code*for*BDD. ...##
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Page 2336 of Mathematical Reviews Vol. , Issue 88e
[page]

1988
*
Mathematical Reviews
*

Several examples of o-

*permut*- able n-*groups*are given, and necessary and sufficient conditions*for*an n-*group*to be o-*permutable*are obtained. ... If T is*a*Markov operator on*a**compact*space X then n++ 7" can be viewed as*a**representation*of the semigroup N in the set M(X) of Markov operators on X. ...##
###
Page 1443 of Mathematical Reviews Vol. , Issue 96c
[page]

1996
*
Mathematical Reviews
*

*for*the

*representations*of

*a*maximal torus. ... In this paper the author establishes

*a*strong link be- 96c:2 2006 tween the order structure

*for*

*representation*rings of

*a*

*compact*semisimple connected Lie

*group*G and the corresponding order structure ...

##
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Arithmetic Design for Permutation Groups
[chapter]

1999
*
Lecture Notes in Computer Science
*

We introduce

doi:10.1007/3-540-48059-5_11
fatcat:wxcsfmakg5ftjpmhzobmi3im4q
*a*non-redundant*representation*of the elements of binary*groups*, the so-called*compact**representation*, which allows lowcost realization of arithmetic*for*binary*groups*of large degrees such ... We present highly optimized multiplier architectures operating directly on the*compact*form of*permutations*. Finally, we give complexity and performance estimations*for*the presented architectures. ... Actually, the n−1=7 control bits can be seen as*a*special*representation*of the*group*elements of H 3 , which we call the*compact**representation*. ...##
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Operations on Hypermaps, and Outer Automorphisms

1988
*
European journal of combinatorics (Print)
*

These are induced by the

doi:10.1016/s0195-6698(88)80052-0
fatcat:i5mwonuy6fcvdbhzhlgfsptdiq
*group*of automorphisms of*a*certain*group*G whose transitive*permutation**representations*correspond to hypermaps. ... We present*a**group*f), isomorphic to PGL(2, Il), of operations on hypermaps, and*a**group*D, isomorphic to GL(2, Il), of operations on oriented hypermaps without boundary. ...*A*. Jones*for*suggesting an investigation of the*group*Out(G). She would also like to thank SERC of the U.K.*for*their financial support during that period of research. ...##
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On the problem of classifying simple compact quantum groups
[article]

2013
*
arXiv
*
pre-print

We review the notion of simple

arXiv:1303.1842v1
fatcat:dzorknxefzd57poskryffdeiom
*compact*quantum*groups*and examples, and discuss the problem of construction and classification of simple*compact*quantum*groups*. ... Several new quantum*groups*constructed by Banica, Curran and Speicher since the author's first paper on simple quantum*groups*are shown to be simple using results of Raum and Weber. ... The author would also like to thank Hanfeng Li and Dechao Zheng at Chongqing University, and Yi-Jun Yao and Guoliang Yu at Fudan University*for*their hospitality and*for*creating*a*congenial atmosphere ...##
###
On the problem of classifying simple compact quantum groups

2012
*
Banach Center Publications
*

We review the notion of simple

doi:10.4064/bc98-0-18
fatcat:nrm4x67bkfa3jnwd64pas7afny
*compact*quantum*groups*and examples, and discuss the problem of construction and classification of simple*compact*quantum*groups*. ... The author would also like to thank Hanfeng Li and Dechao Zheng at Chongqing University, and Yi-Jun Yao and Guoliang Yu at Fudan University*for*their hospitality and*for*creating*a*congenial atmosphere ... The author is indebted to Professor Marek Bożejko*for*drawing his attention to easy quantum*groups*during the conference in September 2011 celebrating Professor Woronowicz's seventieth birthday. ...##
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πDD: A New Decision Diagram for Efficient Problem Solving in Permutation Space
[chapter]

2011
*
Lecture Notes in Computer Science
*

In this paper, we propose

doi:10.1007/978-3-642-21581-0_9
fatcat:4bnebu3rqvenhj7ycg75vgichq
*a*new type of decision diagram named "πDD,"*for**compact*and canonical*representation*of*a*set of*permutations*. ... In addition, πDDs hava*a*special Cartesian product operation which generates all possible composite*permutations**for*two given sets of*permutations*. This is*a*beautiful and powerful property of πDDs. ... This data structure can compress*a*large number of*permutations*into*a**compact*and canonical*representation*. ...##
###
Infinite quantum permutations
[article]

2022
*
arXiv
*
pre-print

We define and study quantum

arXiv:2208.01310v1
fatcat:w4t56aqwi5fonpnpm3gaqk4swa
*permutations*of infinite sets. This leads to discrete quantum*groups*which can be viewed as infinite variants of the quantum*permutation**groups*introduced by Wang. ... More precisely, the resulting quantum*groups*encode universal quantum symmetries of the underlying sets among all discrete quantum*groups*. ... Dually,*a*locally*compact*quantum*group*G is discrete if its dual Ĝ is*compact*. ...##
###
Page 1369 of Mathematical Reviews Vol. 43, Issue 6
[page]

1972
*
Mathematical Reviews
*

*a*

*permutation*of Q

*for*any aeQ. ...

*A*quasigroup Q(-) is called an RIP-quasigroup if there is

*a*

*permutation*7 with yx-7x=y

*for*all x, y €Q. ...

##
###
Page 32 of Mathematical Reviews Vol. , Issue 89A
[page]

1989
*
Mathematical Reviews
*

G, then H € M and G/H € M,;; (ii) if

*A*is*a*set and G, € M*for*each*a*€*A*, then [],- 4 Ga € M; (iii) the class of all*compact*Hausdorff abelian*groups*is*a*proper subclass of M. ... By*a*generalized*permutation*on n symbols the author means*a*mapping f:X —*A*(X), where X = {x,,---,X,}. ...##
###
Page 468 of Mathematical Reviews Vol. 51, Issue 2
[page]

1976
*
Mathematical Reviews
*

*For*instance, Theorem 3: Let G be

*a*locally

*compact*

*group*, U

*a*separable

*compact*subgroup of G, c°

*a*fixed irreducible

*representation*of U; if T is

*a*representa- tion of G homogeneous with respect to U ...

*For*

*a*complex semisimple Lie

*group*G with

*a*maximal

*compact*subgroup U and

*a*unitary

*representation*of G the notion of homogeneity was introduced by the author [Studia Math. 31 (1968), 383-398; MR 38 #4618 ...

##
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Separability criteria via wavelet transform on homogenous spaces and projective representations

2010
*
Open Physics
*

Banach spaces

doi:10.2478/s11534-009-0103-z
fatcat:b3ms2sbgx5bbxh6ymcvqn6l6ae
*for*some well known examples, such as: Moyal*representation**for**a*spin; Dihedral and*Permutation**groups*. ... Here, we present*a*universal description of quantum entanglement using*group*theory and non-commutative characteristic functions*for*homogenous space and projective*representation*of*compact**groups*on ... of the*group*on Banach spaces*for*some of the well known examples, such as: Moyal*representation**for**a*spin, Dihedral and*permutation**groups*. ...##
###
Page 907 of Mathematical Reviews Vol. 18, Issue 11
[page]

1957
*
Mathematical Reviews
*

Let G be

*a*separable locally*compact**group*and let [’) and I's be closed subgroups of G.*For*each unitary repre- sentation L; of [; let U% denote the corresponding in- duced*representation*of G. ... They also show that if S is*a*topo- logical semigroup then any one of the following conditions will imply stability: S is commutative, S is*compact*, S is the union of*groups*, {x"} has*compact*closure*for*...##
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Factor representations and their characters for the wreath products of compact groups with the infinite symmetric group

2005
*
Proceedings of the Symposium on Representation Theory
*

Let T be

doi:10.34508/repsympo.2005.0_121
fatcat:b3zk5elzcnddlj3kcrmc2cyzlu
*a**compact**group*, and denote by G = S ∞ (T ) the wreath product of T with the infinite symmetric*group*S ∞ . (1) We study first characters of facter*representations*of finite type of G and give ...*a*general character formula. (2) Then,*for*each character f , we give*a*realization of*a*factor*representation*π corresponding to it with*a*trace-element v such that f (g) = ⟨π(g)v, v⟩. (3) Further we ... Wreath products of*compact**groups*with the infinite symmetric*group**For**a*set I, we denote by S I the*group*of all finite*permutations*on*A*. ...
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