Download what is the asymptotic theory of repr

Asymptotics of representations of symmetric groups
Piotr Śniady
Plan for today
• context: what is the asymptotic theory of representations of Sn?
example of a problem
• connection coefficients problem: how to deal with products of conjugacy
classes in Sn?
the main combinatorial tool: calculus of partitions
• conclusions from the calculus of partitions
Example of a problem
For integer n ≥ 1 we consider a Young diagram ν with a shape of a n × n
square. A standard Young tableaux is a filling of this Young diagram with
numbers 1, . . . , n2 such that the numbers increase along the diagonals ր,
տ.
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We can think that a Young diagram is a pile of stones and the Young
tableau is the order in which stones are placed.
to be continued. . .
Let 0 < α < 1 be fixed; we remove from a randomly chosen Young tableaux
all boxes with numbers bigger than αn2.
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What is the shape of the resulting Young diagram λ with αn2 boxes, when
n → ∞? What are the statistical properties of a large Young tableaux of a
prescribed shape? In other words: what was the shape of this pile of
stones in the past?
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The above question is a special case of the problem of the structure of
reducible representations of Sn. In this problem we ask: what can we
say about a random Young diagram contributing to a prescribed reducible
representation?
In this particular case we study the restriction of an irreducible representaS 2
ν
tion ρ y n .
Sαn2
What can we learn from the above problem?
• In principle, for any question concerning representations of Sn there is
a well-known answer given by some combinatorial algorithm. However,
when n → ∞, such combinatorial answers are useless. We need more
analytic methods.
• In the asymptotic theory of representations some questions (“what was
the shape of this pile of stones?”) are of statistical flavor. We need
more probabilistic methods.
• Almost every problem in this theory can be reformulated as follows: express
some prescribed central elements in C(Sn) as functions of some other
central elements in C(Sq). For this reason we need an efficient way
to perform calculations on central elements in C(Sn). The rest
of this talk will be devoted to this problem.
For integers k1, . . . , km ≥ 1 we define the conjugacy class Σk1,...,km ∈
C(Sn) as a linear combination of all permutations which in a cycle decomposition have cycles of length k1, . . . , km.
Example:
Σ3,2 =
X
(a1,1, a1,2, a1,3)(a2,1, a2,2),
a1,1 a1,2 a1,3
a2,1 a2,2
where the sum runs over all fillings of the boxes with numbers 1, 2, . . . , n
(every number can appear at most once).
The main technical problem: How to compute explicitly the product
of the conjugacy classes Σk1,...,km · Σk ′ ,...,k ′ ? Up to some simple rescaling
1
m′
this problem is known as the connection coefficients problem.
Solution to this problem: we will index the conjugacy classes by partitions. For a partition π of a set {1, 2, . . . , l} we define
1X
Σπ =
(p1, p2)(p2, p3) · · · (pl−1, pl)(pl, p1) ∈ C(Sn),
n p∼π
where the sum runs over all sequences p1, . . . , pl ∈ {1, . . . , n} such that
pi = pj if and only if i and j are connected by π.
Example: for π = 1 2 3 4 5 6 7 we sum over sequences of the form
(p1, . . . , p7) = (a, b, a, c, b, d, b) with a, b, c, d all different.
Why are Σπ so nice? The answer on the following transparencies. . .
Nice property 1: Products of conjugacy classes can be computed
easily in terms of Σπ.
To be explicit: for any partition π1 of a set {1, . . . , l1} and any partition
π2 of a set {l1 + 1, l1 + 2, . . . , l1 + l2}
X
Σ π1 · Σ π2 =
Σπ ,
π
where the sum runs over all partitions π of the set {1, . . . , l1 + l2} such that
• any elements a, b ∈ {1, . . . , l1} are connected by π if and only if they are
connected by π1,
• any elements a, b ∈ {l1 + 1, . . . , l1 + l2} are connected by π if and only
if they are connected by π2,
• elements l1 and l1 + l2 are connected by π.
Nice property 2: There is an explicit formula for Σπ. We represent
a partition π = 1 2 3 4 5 6 7 on the boundary of a circle, take the boundary
of the picture and make a traffic-circle.
3 2’
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3’
3’
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4’
4’
1’
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5’
5’
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6 6’
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For each loop we count
number of visited vertices
− number of winds ;
2
these numbers specify the conjugacy class of Σπ.
In our example Σπ = Σ 10 −3, 4 −1 = Σ2,1.
2
2
2’
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1’
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7’
Nice property 3: There is a topological interpretation for the
contribution of Σπ
In the asymptotic theory we need to make some approximations; for example
we need to know how big are the characters of the form χρ(Σπ), where ρ is
some nice representation of Sn.
number of vertices of π−2·genus of π
2
χρ(Σπ) = O n
This means: we know which partitions are asymptotically negligible.
Corollaries
There is a very wide class of reducible representations of Sn such that
• (“law of large numbers” Biane, 1998) the contributing Young diagrams
concentrate (in the limit n → ∞) around a prescribed shape;
• (“central limit theorem” Śniady, 2005) the fluctuations of the contributing
Young diagrams around the limit shape are Gaussian (in the limit n → ∞);
• the behavior of the contributing Young diagrams is related to the random
matrix theory and Voiculescu’s free probability theory.
In particular, this gives a complete answer to the problem from the beginning
(“what was the shape of the pile of stones?”).
Advertisement:
• Piotr Śniady, Gaussian fluctuations of characters of symmetric groups and
of Young diagrams. Preprint arXiv:math.CO/0501112
• Piotr Śniady, Asymptotics of characters of symmetric groups and genus
expansion. Preprint arXiv:math.CO/0412043
• abstract to this talk