The Algebra of Conjugacy Classes in Symmetric Groups and Partial Permutations

By V. Ivanov, S. Kerov. 2001

116 citations recorded by the source corpus.

We prove the convolution formula for the conjugacy classes in symmetric groups conjectured by the second author. A combinatorial interpretation of coefficients is provided. As a main tool, we introduce a new semigroup of partial permutations. We describe its structure, representations, and characters. We also discuss filtrations on the subalgebra of invariants in the semigroup algebra. Bibliography: 10 titles.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/08f21a874ba0df00e0fe3f488a48daacddf7ae2b

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In Discrete Mathematics and Combinatorics

1. Piotr Śniady
2. Valentin F'eray
3. Grigori Olshanski
4. Maciej Dolkega
5. A. Mironov
6. A. Morozov
7. V. Ivanov
8. Philippe Biane

In Geometry and Topology

1. A. Morozov
2. Piotr Śniady
3. Grigori Olshanski
4. A. Mironov
5. V. Ivanov
6. S. Natanzon
7. Roland Speicher
8. I. Cherednik

Citations in the corpus

• Kerov’s Central Limit Theorem for the Plancherel Measure on Young Diagrams (2003, methodology): “er on the k-cycle, and Lagrange’s inversion formula. In §4, we introduce the basis {p# ρ }in A and study a family of filtrations in A, which are defined in terms of this basis. Here we follow the paper [IK]. We 4 essentially need two different filtrations. Their purpose is to single out main terms of as”
• Algebra of differential operators associated with Young diagrams (2012, background): “Then, ρ1([1]) = 2[1, 1], ρ2([1]) = 3[1, 1, 1], ρ1([2]) = [2, 1].”
• GAUSSIAN FLUCTUATIONS OF YOUNG DIAGRAMS AND STRUCTURE CONSTANTS OF JACK CHARACTERS (2014, background): “well-known [Sag01, equation (1.2)] to be (k +n)! z (k1n) = k +n k (k −1)!. Remark. The quantities gµ,ν (1) ;π also have a direct combinatorial interpretation in terms of partial permutations, see [IK99]. 5.2. Case α = 2: Hecke algebra of (S 2n,Hn). An analogous interpretation of the structure cons”
• Gaussian fluctuations of characters of symmetric groups and of Young diagrams (2005, background): “s convenient to consider their disjoint product defined by (2.2) Σk 1,...,km •Σl1,...,ln = Σk1,...,km,l1,...,ln. Remark. The readers familiar with the notion of partial permutations of Ivanovand Kerov [IK99] will see that (2.2) is compatiblewith the following 6 PIOTR SNIADY´ definition of the product ”