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Convert between different ways of expressing permutations

asked 2020-06-01 00:56:29 +0200

Peter M gravatar image

updated 2020-06-01 10:46:18 +0200

FrédéricC gravatar image

Suppose I have two permutations x and y, given in one line notation (e.g. x=[2,1,4,3] and y=[3,2,1,4]). How would I go about computing e.g. the Kazhdan-Lusztig polynomial Px,y(q) in sage? The KL polynomial requires as input representing symmetric group elements as products of simple reflections and I do not know how to (using a computer) convert between these different ways of expressing a permutation. How does one do this in sage?

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answered 2020-06-01 02:44:10 +0200

tmonteil gravatar image

I am not sure about your exact question, but you might start with:

sage: x = Permutation([2,1,4,3])
sage: y = Permutation([3,2,1,4])
sage: x.cycle_tuples()
[(1, 2), (3, 4)]
sage: y.cycle_tuples()
[(1, 3), (2,), (4,)]

Regarding Kazhdan-Lusztig polynomial for Weyl groups, you can have a look at :

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answered 2020-06-01 10:39:34 +0200

FrédéricC gravatar image

updated 2020-06-01 10:40:29 +0200

Here is a sample session

sage: S4 = SymmetricGroup(4)
sage: R.<q> = LaurentPolynomialRing(QQ)
sage: KL = KazhdanLusztigPolynomial(S4,q)
sage: x = S4([1,2,3,4])
sage: y = S4([2,3,4,1])
sage: KL.P(x, y)
1
sage: y = S4([4,2,3,1])
sage: KL.P(x, y)
1 + q
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Asked: 2020-06-01 00:56:29 +0200

Seen: 41 times

Last updated: Jun 01 '20