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Let $\lambda,\mu,\nu$ be some partitions. Let's denote with $s_\lambda,s_\mu,s_\nu$ the Schur functions associated to these partitions. If

$s_\mu s_\nu=\sum_\lambda c^\lambda_{\mu\nu}s_\lambda$

then the Schur skew function is

$s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu\nu}s_\nu$

how can I prove that $s_{\lambda/\mu}=\sum_T x^T$ where $T$ is a tableaux of shape $\lambda/\mu$? (so we are supposing that $\mu\subset\lambda$)

(I know that $c^\lambda_{\mu\nu}=0$ if $|\lambda|\neq |\mu|+|\nu|$)

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Do you know the Littlewood-Richardson rule (that tells you what the $c_{\mu\nu}^\lambda$ count) and its proof yet? If yes, then the expression for skew shapes in terms of standard tableaux should follow from that proof and the expression of the regular schur functions in terms of standard tableaux. I myself don't know of a simpler way to get to the result you want, and the proof of the Littlewood-Richardson rule seems sufficiently long and complicated to warrant being looked up (there could exist a simpler proof; my knowledge of this stuff is fairly limited). –  Vladimir Sotirov Jun 26 '11 at 16:50
    
Actually I need that for the proof of the Littlewood-Richardson rule. Because that tell me that $s_{\lambda/\mu}=\sum_\nu K_{\lambda/\mu,\nu}m_\nu$ where the coefficient of $m_\nu$ is the number of tableaux skew of shape $\lambda/\mu$ and weight $\nu$. This implies that $K_{\lambda/\mu,\nu}=<s_{\lambda/\mu},h_\nu>$ that is what I need. –  Jacob Fox Jun 26 '11 at 17:24
    
How do you define your scalar product $\langle s,h \rangle$? –  draks ... Mar 26 '12 at 11:29
    
I had a similar about calculating the Kostka numbers here. What do you think? –  draks ... Mar 30 '12 at 18:46
1  
@draks: The scalar product usually considered for the symmetric functions (and certainly intended here) is the one for which the (straight) Schur functions $s_\lambda$ form an orthonormal basis. For this scalar product the "monomial" functions $m_\lambda$ have as dual basis the complete homogeneous functions $h_\lambda$. –  Marc van Leeuwen Mar 31 '12 at 9:19
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