The symmetric functions catalog

An overview of symmetric functions and related topics

2023-09-20

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

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

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. See also [TG22] for conditions on when $c^{\lambda}_{\mu \nu}$ is positive.

The problem of determining for which $(\lambda, \mu, \nu)$ we have $c^{\lambda}_{\nu\mu} > 0,$ is determined by the Horn inequalities, see [Kly98KT99].

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

Semistandard 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.

Semistandard Young tableaux II

In [FG93MS99], the following model us described. $c^{\lambda}_{\mu\nu}$ is the number of semistandard Young tableaux $T$ with shape $\mu,$ with the property that the column index reading word of $T,$ fits the skew shape $\lambda/\mu.$ That is, we take the column reading word of $T,$ reading columns right to left. This gives a word $w.$ For each entry $w_i,$ we put an entry $i$ in row $w_i,$ so that the resulting filling $F=F(T)$ is of shape $\lambda/\mu.$ If this filling is a skew SYT, then we say that $T$ fits $\lambda/\mu.$

Note that the conditions force that $T$ has content $\lambda-\mu.$

Observe that this model is generalized to Littlewood–Richardson coefficiens for shifted Schur functions.

Example (Fomin–Greene model).

Let us show that $c^{542}_{321,32}=2$ using the Fomin–Greene model. We are counting SSYT with shape $\mu=321,$ which fit $542/32.$

There are two such SSYT (and corresponding SYT obtained from the column reading word):

$1$$1$$2$
$2$$3$ 
$3$  
   $2$$4$
  $1$$5$ 
$3$$6$   
$1$$1$$3$
$2$$2$ 
$3$  
   $2$$4$
  $3$$5$ 
$1$$6$   

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

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 [BZ92] gave a combinatorial model for the Littlewood–Richardson coefficients that uses certain triangular arrays, reminiscent of Gelfand–Tsetlin patterns. The exact definition is written in an accessible mannar in [Def. A1.3.10, Sta01].

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

W. Wen 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, see [Wen21].

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 recent papers on these coefficients, see [GOY20aGOY20b].

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 [GOY20a].

In [Thm. 5.1, GOY20a], a polytope description for $N_{\mu,\nu,\lambda}$ is given, and a saturation-type conjecture is stated. See also [Kin21] for more conjectures.

Stretched Littlewood–Richardson coefficients

The Littlewood–Richardson coefficients are in correspondence with lattice points in certain rational polytopes, as proved in [BZ88]. Consequently, it is immediate that this function is a quasi-polynomial, since we can model this as the Ehrhart (quasi)polynomial of a rational polytope. Polynomiality was later proved by H. Derksen and J. Weyman [DW02], and E. Rassart [Ras04a]. A simple proof of polynomiality is given in [Tha22]. The proof uses the Kostant partition function formula due to Steinberg.

Conjecture (See [KTT04]).

The function

\[ t \to c^{t \nu}_{t\lambda,t\mu} \]

for fixed partitions, is a polynomial in $t.$ Note that this is the Ehrhart polynomial of the corresponding BZ-polytope (where lattice points are BZ-patterns).

Computer experiments, see [KTT04], suggests that the Ehrhart polynomial above has non-negative coefficients. This conjecture is still open. A special case of this conjecture is to consider the map $t \to K_{t\lambda,t\mu},$ of so called stretched Kostka coefficients. Various aspects of this problem has been considered in [Ale16aAle19b].

Fulton's conjecture stated that $c^{\nu}_{\lambda,\mu}=1$ implies that $c^{t \nu}_{t\lambda,t\mu}=1$ for all $t \geq 1.$ This was later proved in [KTW04]. The next natural case is, $c^{\nu}_{\lambda,\mu}=2$ and it is proved by C. Ikenmenyer [Ike16] and later C. Sherman [She15She17] that

\[ c^{\nu}_{\lambda,\mu}=2 \implies c^{t \nu}_{t\lambda,t\mu}=t+1. \]

A nice generalization of this implication is given in [CR22a], where it is proved that

\[ c^{\nu}_{\lambda,\mu}=2 \implies c^{\nu(s,t)}_{\lambda(s,t),\mu(s,t)}=\binom{s+t}{t}, \]

with the convention that $\lambda(s,t) \coloneqq s (t \lambda')'$ (i.e., we stretch in both vertical and horizontal direction).

Stretching of Newell–Littlewood coefficients is studied in [Kin21]. This preprint conttains lots of interesting conjectures analogous to results/conjectures about the Littlewood–Richardson coefficients. R. C. King proves that the map $t \to n^{t \nu}_{t\lambda,t\mu}$ is a quasipolynomial with period at most 2.

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

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

Shrivastava's relation with Kostka numbers

In [Shr22], Shrivastava shows how one can compute Littlewood–Richardson coefficients as a signed sum of Kostka coefficients.

It is known that (skew) Kostka coefficients can be obtained as a special case of Littlewood–Richardson coefficients, see e.g. [p. 338, Eq. 7.61, Sta01].

Computing coefficients, and A. Buch's lrcalc

H. Narayanan proved in 2006 that computing Littlewood–Richardson coefficients and Kostka coefficients is in general $\#P$-complete, see [Nar06]. This implies that there are no efficient algorithms for computing these quantities (assuming wildly believed conjectures regarding complexity theory).

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

See also Matthew J. Samuel's python package for computing products of Schubert polynomials which, in particular, can compute Littlewood–Richardson coefficients.

References