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Skew Schur function

asked 2022-04-17 06:23:20 +0200

Anupamsage gravatar image

I have worked with the Schur function using the following command for example

sage: Sym = SymmetricFunctions(QQ)
sage: P = Sym.p()  # to expand symmetric functions in power symmetric basis
sage: s = Sym.schur()
sage: P(s([2,2]))
1/12*p[1, 1, 1, 1] + 1/4*p[2, 2] + (-1/3)*p[3, 1]

Recently I need to work with the skew Schur function. I want to expand the shifted schur function $$ s_{\lambda}(x_1 +y , x_2 +y ,\ldots , x_n +y) $$ I am guessing that it can be the following identity $$ s_{\lambda}(x_1 +y , x_2 +y ,\ldots , x_n +y) = \sum_{\mu} s_{\lambda \setminus mu}(y) s_{\mu}(x_1 , x_2, \ldots, x_n) $$ For $y=1$ my idenity can be proven as I am sure there is an expression. The expression $s_{\lambda \setminus mu}(y)$ is skew Schur function. I am wondering if there are packages in sagemath where I can work with skew schur function and verify my identity is correct or not.

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answered 2022-04-17 13:55:52 +0200

FrédéricC gravatar image

Skew Schur functions are available:

            sage: sp = SkewPartition([[5,3,3,1], [3,2,1]])
            sage: s = SymmetricFunctions(QQ).s()
            sage: s.skew_schur(sp)
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Thanks a lot

Anupamsage gravatar imageAnupamsage ( 2022-04-18 07:39:13 +0200 )edit

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Asked: 2022-04-17 06:23:20 +0200

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Last updated: Apr 17 '22