The symmetric functions catalog

An overview of symmetric functions and related topics

2022-07-29

Combinatorial models for Littlewood–Richardson coefficients

The Littlewood–Richardson coefficients are defined as the structure constants $c^{\lambda}_{\mu\nu}$ appearing in the expansion $\schurS_\mu \schurS_\nu = \sum_\lambda c^{\lambda}_{\mu\nu} \schurS_\lambda.$ One can show that $c^{\lambda}_{\mu\nu}=0$ unless $\mu,\nu \subseteq \lambda.$ Furthermore, we have the symmetries

\[ c^{\lambda}_{\mu\nu} = c^{\lambda}_{\nu\mu} = c^{\lambda'}_{\mu'\nu'}. \]

Hariharan Narayanan proved in 2006 that it is $\#P$-complete to compute $c^{\lambda}_{\mu\nu}.$

A simple recursion for computing $c^{\lambda}_{\nu\mu}$ can be obtained from the shifted Schur functions.

In [Aze99], it is proved that for fixed $\lambda$ and $\mu,$ the set

\[ \{ \nu : c^{\lambda}_{\mu \nu}>0 \} \]

is a sub-lattice of the partition lattice under domination order.

Littlewood–Richardson tableaux

The Littlewood–Richardson rule states that $c^{\lambda}_{\mu\nu}$ count the number of skew semi-standard Young tableaux of shape $\lambda/\mu$ with content $\nu,$ with the additional restriction that the concatenation of the reversed rows is a lattice word. A word $w_1,\dotsc,w_\ell$ is a lattice word if every prefix has the property that its content is a partition. That is, the number of times $i$ appears in the prefix is at least as many as $i+1,$ for every $i.$ Tableaux of this type are called Littlewood–Richardson tableaux.

Example (Littlewood–Richardson tableaux).

Consider the coefficient of $\schurS_{542}$ in the product $\schurS_{32}\cdot \schurS_{321}.$ We must find all Littlewood–Richardson tableaux of shape $542/32$ with content $321.$ There are two such tableaux:

   $1$$1$
  $1$$2$ 
$2$$3$   
   $1$$1$
  $2$$2$ 
$1$$3$   

Consequently, $c^{542}_{32,321}=2.$

Standard Young tableaux

In [Cor. 5.4.8 , Lot02], they give the following characterization of Littlewood–Richardson coefficients: $c^{\lambda}_{\mu\nu}$ is the number of semistandard Young tableaux $T$ with shape $\nu,$ and content $\lambda-\mu$ such that $\rw(t) \cdot \ell^{\mu_\ell} \dotsm 2^{\mu_2}1^{\mu_1}$ is a Yamanouchi word.

By using [Rem. 14 and Cor. 21, KM19], one can show that this is equivalent to the following model.

Postfix-charge model

In [AU20] we show that

\[ c^{\lambda}_{\mu\nu} = \left| T \in \SSYT(\nu, \lambda - \mu) : \charge( \rw(T) \cdot \ell^{\mu_\ell} \dotsm 2^{\mu_2}1^{\mu_1})=0 \right|. \]

That is, $c^{\lambda}_{\mu\nu}$ is equal to the number of semistandard Young tableaux of shape $\nu$ and content $\lambda - \mu$ such that the reading word postfixed with $\ell^{\mu_\ell} \dotsm 2^{\mu_2}1^{\mu_1}$ has charge equal to zero.

Example (Postfix-charge model).

Let us compute $c^{33221}_{221,321},$ where $\lambda = 33221,$ $\mu = 221$ and $\nu = 321.$ We are interested in semistandard tableaux with shape $321$ and content $\lambda-\mu = 11121.$ There are eight such tableaux.

$1$$4$$4$
$2$$5$ 
$3$  
$1$$3$$5$
$2$$4$ 
$4$  
$1$$3$$4$
$2$$5$ 
$4$  
$1$$3$$4$
$2$$4$ 
$5$  
$1$$2$$5$
$3$$4$ 
$4$  
$1$$2$$4$
$3$$5$ 
$4$  
$1$$2$$4$
$3$$4$ 
$5$  
$1$$2$$3$
$4$$4$ 
$5$  

We take their reading words and concatenate $\ell^{\mu_\ell} \dotsm 2^{\mu_2}1^{\mu_1} = 32211$ at the end.

$\text{Word}$$\text{Charge}$
$32514432211$$1$
$42413532211$$1$
$42513432211$$0$
$52413432211$$0$
$43412532211$$2$
$43512432211$$1$
$53412432211$$2$
$54412332211$$1$
As we see, the third and fourth tableau above give charge $0,$ so $c^{33221}_{221,321}=2.$

Example (Postfix-charge model II).

Let us compute $c^{542}_{32,321} = c^{\lambda}_{\mu\nu}.$ We are interested in semistandard tableaux with shape $321$ and content $\lambda-\mu = 222.$ There are two such tableaux.

$1$$1$$3$
$2$$2$ 
$3$  
$1$$1$$2$
$2$$3$ 
$3$  

We take their reading words and concatenate $22111$ at the end, $322113\;22111$ and $323112\;22111.$ Both of these words have charge $0,$ so $c^{542}_{32,321}=2.$

Berenstein–Zelevinsky patterns

Berenstein and Zelevinsky [BZ88] gave a combinatorial model for the Littlewood–Richardson coefficients that uses certain triangular arrays, reminiscent of Gelfand–Tsetlin patterns.

Knutson–Tao honeycombs

Knutson–Tao–Woodward puzzles

Knutson–Tao–Woodward puzzles [KT99KTW04] is a different model used for proving the saturation conjecture. A nice overview of the saturation conjecture is given by A. Buch [Buc00].

Steinberg's formula

One can express the Littlewood–Richardson coefficients as a signed sum involving the Kostant partition function, by using Steinberg's formula [Ste61].

See the exact formula in the section with the Littlewood–Richardson coefficiens from the Kostant partition function.

Socle tableaux

In [KS], $c^{\lambda}_{\mu\nu}$ is realized as the number of socle tableaux of shape $\lambda/\nu$ and content $\mu.$ Note that the role of $\mu$ and $\nu$ have been interchanged.

A socle tableau is weakly decreasing in rows, strictly decreasing in columns, and for each vertical line and integer $\ell,$ there are at most as many entries $\ell+1$ on the left hand side, as entries $\ell.$

Evaluating Schur functions at roots of unity

In https://arxiv.org/pdf/2107.09907.pdf, the author gives a fairly explicit formula for Littlewood–Richardson coefficients, as a sum over partitions, where the terms include a product of three Schur functions, evaluated at certain roots of unity.

Newell–Littlewood numbers

The Newel–Littlewood numbers may be defined as

\[ N_{\mu,\nu,\lambda} = \sum_{\alpha,\beta,\gamma} c^{\mu}_{\alpha\beta} c^{\nu}_{\alpha\gamma} c^{\lambda}_{\beta\gamma}. \]

For a recent paper on these coefficients, see [GOY20].

The Littlewood–Richardson coefficients arise as tensor product multiplicities of irreducible representations of $\GL(V).$ The Newel–Littlewood numbers arise in the same manner, where $\GL(V)$ is replaced by a Lie group of type $B,$ $C$ or $D.$

There is a symmetric function basis, the Koike–Terada basis, $\{\schurS_{[\lambda]}\}_\lambda$ for which the $N_{\mu,\nu,\lambda}$ serve as multiplication structure constants. An explicit Jacobi–Trudi type formula for this basis is given in [GOY20].

In [Thm. 5.1, GOY20], a polytope description for $N_{\mu,\nu,\lambda}$ is given, and a saturation-type conjecture is stated.

A. Buch's lrcalc

Anders Buch has a fast C program, lrcalc, for computing Littlewood–Richardson coefficients.

Identities

Coquereaux–Zuber relation

Let $\mu, \nu \vdash n$ be such that all entries are at most $k.$ In [CZ], it is proved that

\[ \sum_{\lambda \vdash n} c^{\lambda}_{\mu, \nu} = \sum_{\lambda \vdash n} c^{\lambda}_{\mu^{\vee k}, \nu}, \]

where $\mu^{\vee k} \coloneqq (k-\lambda_n,\dotsc,k-\lambda_2,k-\lambda_1).$

The Harris–Willenbring identity

See https://link.springer.com/chapter/10.1007/978-1-4939-1590-3_11

\[ \sum_{n \geq 0} \sum_{\lambda,\mu, \nu \vdash n} \left( c^{\lambda}_{\mu\nu} \right)^2 t^{|\mu|}q^{|\nu|} = \prod_{i \geq 1} \frac{1}{1-t^i-q^i} \]

References