Littlewood–Richardson rule

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two <a href="/facts/Schur_polynomial/mBzF2Mdr">Schur functions</a> as a linear combination of other Schur functions. These coefficients are natural numbers, which the Littlewood–Richardson rule describes as counting certain <a href="/facts/Skew_tableau/e5lI4bck">skew tableaux</a>. They occur in many other mathematical contexts, for instance as <a href="/facts/Multiplicity_(mathematics)/dndUyetA">multiplicity</a> in the decomposition of <a href="/facts/Tensor_product/7K3pR1ap">tensor products</a> of <a href="/facts/Representations_of_classical_Lie_groups/BfBTWQYz">finite-dimensional representations of general linear groups</a>, or in the decomposition of certain <a href="/facts/Induced_representations/Jh25PGDP">induced representations</a> in the <a href="/facts/Representation_theory_of_the_symmetric_group/ht0T7JNt">representation theory of the symmetric group</a>, or in the area of <a href="/facts/Algebraic_combinatorics/9b47n3Fh">algebraic combinatorics</a> dealing with <a href="/facts/Young_tableaux/e5lI4bck">Young tableaux</a> and <a href="/facts/Symmetric_polynomials/GNhguVcn">symmetric polynomials</a>.
Littlewood–Richardson coefficients depend on three <a href="/facts/Integer_partition/LtyXWqwH">partitions</a>, say 
 
 
 
 λ
 ,
 μ
 ,
 ν
 
 
 {\displaystyle \lambda ,\mu ,\nu }
 
, of which 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 and 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 describe the Schur functions being multiplied, and 
 
 
 
 ν
 
 
 {\displaystyle \nu }
 
 gives the Schur function of which this is the coefficient in the linear combination; in other words they are the coefficients 
 
 
 
 
 c
 
 λ
 ,
 μ
 
 
 ν
 
 
 
 
 {\displaystyle c_{\lambda ,\mu }^{\nu }}
 
 such that

s
          
            λ
          
        
        
          s
          
            μ
          
        
        =
        
          ∑
          
            ν
          
        
        
          c
          
            λ
            ,
            μ
          
          
            ν
          
        
        
          s
          
            ν
          
        
        .
      
    
    {\displaystyle s_{\lambda }s_{\mu }=\sum _{\nu }c_{\lambda ,\mu }^{\nu }s_{\nu }.}

The Littlewood–Richardson rule states that 
 
 
 
 
 c
 
 λ
 ,
 μ
 
 
 ν
 
 
 
 
 {\displaystyle c_{\lambda ,\mu }^{\nu }}
 
 is equal to the number of Littlewood–Richardson tableaux of <a href="/facts/Skew_tableau/e5lI4bck">skew shape</a> 
 
 
 
 ν
 
 /
 
 λ
 
 
 {\displaystyle \nu /\lambda }
 
 and of weight 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
.

Littlewood–Richardson rule open-in-new

Littlewood–Richardson rule