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.
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.
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.
$\text{Word}$ | $\text{Charge}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
$32514432211$ | $1$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
$42413532211$ | $1$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
$42513432211$ | $0$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
$52413432211$ | $0$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
$43412532211$ | $2$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
$43512432211$ | $1$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
$53412432211$ | $2$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
$54412332211$ | $1$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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
- [AU20] Per Alexandersson and Joakim Uhlin. Cyclic sieving, skew Macdonald polynomials and Schur positivity. Algebraic Combinatorics, 3(4):913-939, 2020.
- [Aze99] Olga Azenhas. The admissible interval for the invariant factors of a product of matrices. Linear and Multilinear Algebra, 46(1-2):51–99, July 1999.
- [Buc00] Anders Skovsted Buch. The saturation conjecture (after A. Knutson and T. Tao), with an appendix by William Fulton. Enseign. Math. (2), 46:43–60, 2000.
- [BZ88] A. D. Berenstein and A. V. Zelevinsky. Tensor product multiplicities and convex polytopes in partition space. Journal of Geometry and Physics, 5(3):453–472, 1988.
- [CZ] Robert Coquereaux and Jean-Bernard Zuber. On sums of tensor and fusion multiplicities. Journal of Physics A: Mathematical and Theoretical, 44(29):295208, June .
- [GOY20] Shiliang Gao, Gidon Orelowitz and Alexander Yong. Newell–Littlewood numbers. arXiv e-prints, 2020.
- [KM19] Ryan Kaliszewski and Jennifer Morse. Colorful combinatorics and Macdonald polynomials. European Journal of Combinatorics, 81:354–377, October 2019.
- [KS] Justyna Kosakowska and Markus Schmidmeier. The socle tableau as a dual version of the Littlewood–Richardson tableau. Journal of the London Mathematical Society, May .
- [KT99] Allen Knutson and Terence Tao. The honeycomb model of $GL_n(C)$ tensor products I: proof of the saturation conjecture. Journal of the American Mathematical Society, 12(04):1055–1091, October 1999.
- [KTW04] Allen Knutson, Terence Tao and Christopher Woodward. The honeycomb model of $GL_n(C)$ tensor products II: puzzles determine facets of the Littlewood–Richardson cone. Journal of the American Mathematical Society, 17(1):19–49, January 2004.
- [Lot02] M. Lothaire. Algebraic combinatorics on words. Cambridge University Press, 2002.
- [Ste61] Robert Steinberg. A general Clebsch–Gordan theorem. Bull. Amer. Math. Soc., 67(4):406–407, July 1961.