Presentations for (singular) partition monoids: a new approach

East, James

doi:10.1017/s030500411700069x

Cited by 7 publications

(5 citation statements)

References 26 publications

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“…It also follows from some of the above identities. We will also require the following result from [37]; see also [44,46]. For 1 ≤ i ≤ n, let π i ∈ P n be the projection with domain [n] \ {i} and kernel ∆.…”

Section: The Partition Monoidmentioning

confidence: 99%

“…The original study of cellular semigroup algebras may be found in [35]; see also [73,74,112,113] for some recent developments. Another example of how the study of these semigroups can give information about the associated algebras may be found in work of the first author [36], who gives presentations for the partition monoid and shows how these presentations give rise to presentations for the partition algebra; see also [37,43,44]. A further example is given in the paper [27] where idempotents in the partition, Brauer and partial Brauer monoids are described and enumerated, and then the results are applied to determine the number of idempotent basis elements in the finite dimensional partition, Brauer and partial Brauer algebras; see also [28].…”

Section: Introductionmentioning

confidence: 99%

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Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals

East

Gray

2017

Journal of Combinatorial Theory, Series A

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Abstract. We study the ideals of the partition, Brauer, and Jones monoid, establishing various combinatorial results on generating sets and idempotent generating sets via an analysis of their Graham-Houghton graphs. We show that each proper ideal of the partition monoid Pn is an idempotent generated semigroup, and obtain a formula for the minimal number of elements (and the minimal number of idempotent elements) needed to generate these semigroups. In particular, we show that these two numbers, which are called the rank and idempotent rank (respectively) of the semigroup, are equal to each other, and we characterize the generating sets of this minimal cardinality. We also characterize and enumerate the minimal idempotent generating sets for the largest proper ideal of Pn, which coincides with the singular part of Pn. Analogous results are proved for the ideals of the Brauer and Jones monoids; in each case, the rank and idempotent rank turn out to be equal, and all the minimal generating sets are described. We also show how the rank and idempotent rank results obtained, when applied to the corresponding twisted semigroup algebras (the partition, Brauer, and Temperley-Lieb algebras), allow one to recover formulae for the dimensions of their cell modules (viewed as cellular algebras) which, in the semisimple case, are formulae for the dimensions of the irreducible representations of the algebras. As well as being of algebraic interest, our results relate to several well-studied topics in graph theory including the problem of counting perfect matchings (which relates to the problem of computing permanents of {0, 1}-matrices and the theory of Pfaffian orientations), and the problem of finding factorizations of Johnson graphs. Our results also bring together several well-known number sequences such as Stirling, Bell, Catalan and Fibonacci numbers.

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Section: The Partition Monoidmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals

East

Gray

2017

Journal of Combinatorial Theory, Series A

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“…For Assumption 2, we take the partitions λ n ∈ P n,n+1 and ρ n ∈ P n+1,n shown in Figure 3; clearly λ n ρ n = ι n for all n ∈ N. For Assumption 3, we require presentations for the partition monoids P n (n ∈ N). Such presentations are stated in [40]; proofs may be found in [29,30]. First, for n ∈ N define an alphabet…”

Section: The Partition Categorymentioning

confidence: 99%

“…Theorem 3.1 (cf. [29,30,40]). For any n ∈ N, the partition monoid P n has presentation X n : R n via φ n .…”

Section: The Partition Categorymentioning

confidence: 99%

“…Presentations for the partition algebras were given by Halverson and Ram in [40], along with a sketch of a proof; full proofs may be found in [29,30]. The techniques used in [29,30] are semigrouptheoretic in nature; results concerning the algebras are deduced from corresponding results on associated diagram monoids, via the theory of twisted semigroup algebras [77]. Presentations for other diagram monoids have been given by other authors; see for example [14,41,55,62].…”

Section: Introductionmentioning

confidence: 99%

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Presentations for tensor categories

East¹

2020

Preprint

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Partition Algebras and the Invariant Theory of the Symmetric Group

Benkart

Halverson

2019

Association for Women in Mathematics Series

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The symmetric group S n and the partition algebra P k (n) centralize one another in their actions on the k-fold tensor power M ⊗k n of the n-dimensional permutation module M n of S n . The duality afforded by the commuting actions determines an algebra homomorphism Φ k,n : P k (n) → End S n (M ⊗k n ) from the partition algebra to the centralizer algebra End S n (M ⊗k n ), which is a surjection for all k, n ∈ Z ≥1 , and an isomorphism when n ≥ 2k. We present results that can be derived from the duality between S n and P k (n); for example, (i) expressions for the multiplicities of the irreducible S n -summands of M ⊗k n , (ii) formulas for the dimensions of the irreducible modules for the centralizer algebra End S n (M ⊗k n ), (iii) a bijection between vacillating tableaux and set-partition tableaux, (iv) identities relating Stirling numbers of the second kind and the number of fixed points of permutations, and (v) character values for the partition algebra P k (n). When 2k > n, the map Φ k,n has a nontrivial kernel which is generated as a two-sided ideal by a single idempotent. We describe the kernel and image of Φ k,n in terms of the orbit basis of P k (n) and explain how the surjection Φ k,n can also be used to obtain the fundamental theorems of invariant theory for the symmetric group.

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Presentations for (singular) partition monoids: a new approach

Abstract: We give new, short proofs of the presentations for the partition monoid and its singular ideal originally given in the author's 2011 papers in J. Alg. and I.J.A.C.

Cited by 7 publications

References 26 publications

Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals

Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals

Presentations for tensor categories

Partition Algebras and the Invariant Theory of the Symmetric Group

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