#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.
The Littlewood–Richardson coefficients generalize the skew Kostka coefficients. I. Pak and E. Vallejo compare several standard models for these coefficients as integer points in Littlewood–Richardson cones [PV05]. They give explicit linear bijections between Littlewood–Richardson tableaux, Knutson–Tao hives, and Berenstein–Zelevinsky triangles, showing that these models have the same underlying combinatorial structure. The hives and BZ-polytopes page collects the polyhedral viewpoint, saturation theorem, and Ehrhart stretching questions.
Jeu-de-taquin gives a formulation using standard tableaux. If \(U\) is a fixed standard Young tableau of shape \(\nu,\) then \(c^\lambda_{\mu\nu}\) is the number of standard tableaux of skew shape \(\lambda/\mu\) whose rectification is \(U\) [Ful97, Sec. 5]. This count is independent of the chosen standard tableau \(U\) of shape \(\nu.\)
O. Azenhas proves that for fixed \(\lambda\) and \(\mu,\) the set \[\{ \nu : c^{\lambda}_{\mu \nu}\gt{}0 \}\] is a sub-lattice of the partition lattice under domination order [Aze99]. See also [TG22] for conditions on when \(c^{\lambda}_{\mu \nu}\) is positive.
The Horn inequalities determine for which triples \((\lambda, \mu, \nu)\) we have \(c^{\lambda}_{\nu\mu} \gt{} 0\); see [Kly98, KT99].
#Littlewood–Richardson tableaux
The Littlewood–Richardson rule states that \(c^{\lambda}_{\mu\nu}\) counts the number of skew semistandard 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:
Consequently, \(c^{542}_{32,321}=2.\)
#Semistandard Young tableaux
Lothaire gives the following characterization of Littlewood–Richardson coefficients [Lot02, Cor. 5.4.8]: \(c^{\lambda}_{\mu\nu}\) 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. Here Yamanouchi word means that every prefix has partition content.
Using [KM19, Rem. 14 and Cor. 21], one can show that this is equivalent to the following model.
#Semistandard Young tableaux II
S. Fomin, C. Greene, A. Molev, and B. Sagan describe the following model [FG93, MS99]. \(c^{\lambda}_{\mu\nu}\) is the number of semistandard Young tableaux \(T\) with shape \(\mu\) and 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 coefficients for shifted Schur functions.
C. Nguyen, S. Nguyen, and D. Woodruff give a formula for Littlewood–Richardson coefficients using peelable tableaux compatible with shuffle tableaux [NNW25]. Their rule behaves well with Bender–Knuth involutions, recovers the usual symmetry of Littlewood–Richardson coefficients, and proves a special case of the Lam–Postnikov–Pylyavskyy Schur log-concavity conjecture.
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, together with the corresponding SYT obtained from the column reading word:
Consequently, \(c^{542}_{32,321}=2.\)
#Postfix-charge model
In joint work with J. Uhlin, we show [AU20] 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.
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 tableaux 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.
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
A. Berenstein and A. 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 manner in [Sta01, Def. A1.3.10].
#Knutson–Tao honeycombs
#Knutson–Tao–Woodward puzzles
Knutson–Tao–Woodward puzzles [KT99, KTW04] are another model used for proving the saturation conjecture. An overview of the saturation conjecture is given by A. Buch [Buc00].
C. Gaetz and Y. Gao relate Schubert puzzles to interlacing triangular arrays [GG25]. They prove that certain rank-three interlacing arrays are equinumerous with proper colorings of a triangular grid graph via generalized Knutson–Tao puzzles, and use the same objects to compute structure constants in \(K(\operatorname{Gr}(d,n)),\) in the localized cohomology of cotangent bundles of Grassmannians, and in the cohomology of partial flag varieties.
P. Anderson extends triangular Schubert puzzles to puzzles with convex polygonal boundary shapes [And24]. The resulting four-, five-, and six-sided puzzles have Schubert-calculus interpretations, give commutativity statements for several symmetric boundary conditions, and express the associated structure constants in terms of Littlewood–Richardson numbers.
E. Briand and M. Rosas study the full group of symmetries of the Littlewood–Richardson coefficients for \(SL_3\) [BR20]. M. Pelletier and N. Ressayre exhibit an unexpected hidden symmetry of Littlewood–Richardson coefficients [PR22]; a proof is given by D. Grinberg [Gri21]. O. Azenhas, A. Conflitti, and R. Mamede describe a faithful action of \(\setZ_2\times D_3\) on Littlewood–Richardson tableaux, companion tableaux, hives, and puzzles [ACM25]. This gives a uniform way to relate commutativity, conjugation, and cyclic symmetries of Littlewood–Richardson coefficients. D. E. Speyer proves the conjecture of T. Lam, A. Postnikov, and P. Pylyavskyy on Schur nonnegativity of certain differences \(\schurS_{\lambda'}\schurS_{\mu'}-\schurS_\lambda\schurS_\mu\) [Spe26]. The proof introduces a new model for Littlewood–Richardson coefficients, called skeps, and proves an L-log-concavity theorem using Murota’s L-convexity.
The skew-hive extension of this model is described on the hive-polytopes page [LN26].
I. Pak and D. Soskin study the largest coefficient \[C(n)=\max_{\substack{|\lambda|=n\\|\mu|+|\nu|=n}} c^\lambda_{\mu,\nu}.\] They prove that at least one maximum is attained with \(\mu\subseteq\nu\subseteq\lambda\) and with \(\nu/\mu\) a disjoint union of single boxes [PS26]. They conjecture that every maximum has this form for \(n\geq16,\) and verify the conjecture computationally through \(n=45.\)
#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 coefficients from the Kostant partition function.
#Socle tableaux
J. Kosakowska and M. Schmidmeier realize \(c^{\lambda}_{\mu\nu}\) as the number of socle tableaux of shape \(\lambda/\nu\) and content \(\mu\) [KS22]. Note that the roles 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 \(\ell,\) at most as many entries \(\ell+1\) occur on the left-hand side as entries \(\ell.\)
#Evaluating Schur functions at roots of unity
X. 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 Newell–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 [GOY20, GOY20].
The Littlewood–Richardson coefficients arise as tensor product multiplicities of irreducible representations of \(\GL(V).\) The Newell–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. S. Gao, G. Orelowitz, and A. Yong give an explicit Jacobi–Trudi type formula for this basis [GOY20].
They also give a polytope description for \(N_{\mu,\nu,\lambda}\) and state a saturation-type conjecture [GOY20, Thm. 5.1]. See also [Kin21] for more conjectures. The saturation conjecture for the Newell–Littlewood numbers has since been proved in [Min24].
#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 quasipolynomial, since we can model it as the Ehrhart quasipolynomial of a rational polytope. Polynomiality was later proved by H. Derksen and J. Weyman [DW02], and by E. Rassart [Ras04]. W. Thawinrak gives a simple proof of polynomiality [Tha22]. The proof uses the Kostant partition function formula due to R. 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 in [KTT04] suggest that the Ehrhart polynomial above has nonnegative 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 have been considered in [Ale16, Ale19].
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].
Mrigendra Singh Kushwaha, K. N. Raghavan, and Sankaran Viswanath study refined Littlewood–Richardson coefficients attached to key polynomials [KRV21]. If \(\chi_{w,\mu}\) is a Demazure character, then the coefficient \(c^\nu_{\lambda\mu}(w)\) is defined by \[T_{w_0}\left(x^\lambda \chi_{w,\mu}(\xvec)\right) = \sum_\nu c^\nu_{\lambda\mu}(w)\schurS_\nu(\xvec).\] They give a hive model for these coefficients and prove a saturation theorem when \(w\) is \(312\)-avoiding, \(231\)-avoiding, or a commuting product of such permutations, generalizing the Knutson–Tao saturation theorem.
The next natural case is \(c^{\nu}_{\lambda,\mu}=2.\) It was proved by C. Ikenmeyer [Ike16] and later by C. Sherman [She15, She17] that \[c^{\nu}_{\lambda,\mu}=2 \implies c^{t \nu}_{t\lambda,t\mu}=t+1.\]
P.-E. Chaput and N. Ressayre give a nice generalization of this implication [CR25]: \[c^{\nu}_{\lambda,\mu}=2 \implies c^{\nu(s,t)}_{\lambda(s,t),\mu(s,t)}=\binom{s+t}{t},\] with the convention that \(\lambda(s,t) \coloneqq s (t \lambda')'\); that is, we stretch in both the vertical and horizontal directions.
Stretching of Newell–Littlewood coefficients is studied in [Kin21]. This preprint contains many conjectures analogous to results and 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 \(k.\) R. Coquereaux and J.-B. Zuber prove that \[\sum_{\lambda \vdash n} c^{\lambda}_{\mu, \nu} = \sum_{\lambda \vdash n} c^{\lambda}_{\mu^{\vee k}, \nu},\] where \(\mu^{\vee k} \coloneqq (k-\lambda_n,\dotsc,k-\lambda_2,k-\lambda_1)\) [CZ11].
#The Harris–Willenbring identity
See [HW14].
\[\sum_{n \geq 0} \sum_{\lambda,\mu, \nu \vdash n} \left( c^{\lambda}_{\mu\nu} \right)^2 t^{|\mu|}q^{|\nu|} = \prod_{i \geq 1} \frac{1}{1-t^i-q^i}\]
#Shrivastava’s relation with Kostka numbers
S. Shrivastava shows how one can compute Littlewood–Richardson coefficients as a signed sum of Kostka coefficients [Shr22].
It is known that (skew) Kostka coefficients can be obtained as a special case of Littlewood–Richardson coefficients; see, for example, [Sta01, p. 338, Eq. 7.61].
#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 [Nar06]. This rules out efficient algorithms for computing these quantities (assuming widely believed conjectures regarding complexity theory).
Peter Bürgisser and C. Ikenmeyer give a polynomial-time algorithm for deciding whether \(c^\nu_{\lambda,\mu}\gt{}0\) [BI13]. Starting from the Knutson–Tao hive model, they interpret \(c^\nu_{\lambda,\mu}\) as the number of capacity-achieving hive flows on the honeycomb graph and obtain an \(O(n^3\log \nu_1)\) positivity test.
A. Buch has a fast C program, lrcalc, for computing
Littlewood–Richardson coefficients [Buc].
See also M. J. Samuel’s schubmult Python package for
computing products of Schubert polynomials, which in
particular can compute Littlewood–Richardson coefficients
[Sam].
Bibliography
- [Ale16]Per Alexandersson. Gelfand–Tsetlin polytopes and the integer decomposition property. European Journal of Combinatorics, 54:1–20, May 2016.
.bib
@article{Alexandersson2016GTPoly, author = {Per Alexandersson}, title = {{G}elfand--{T}setlin polytopes and the integer decomposition property}, journal = {European Journal of Combinatorics}, month = may, publisher = {Elsevier {BV}}, volume = {54}, pages = {1--20}, year = {2016}, issn = {0195-6698}, doi = {10.1016/j.ejc.2015.11.006} } - [Ale19]Per Alexandersson. Polytopes and large counterexamples. Experimental Mathematics, 28(1):115–120, 2019.
.bib
@article{Alexandersson2019CounterExamples, author = {Per Alexandersson}, title = {Polytopes and Large Counterexamples}, journal = {Experimental Mathematics}, year = {2019}, volume = {28}, number = {1}, pages = {115--120}, publisher = {Taylor \& Francis}, doi = {10.1080/10586458.2017.1361367}, URL2 = {http://www.tandfonline.com/doi/abs/10.1080/10586458.2017.1361367} } - [AU20]Per Alexandersson and Joakim Uhlin. Cyclic sieving, skew Macdonald polynomials and Schur positivity. Algebraic Combinatorics, 3(4):913–939, 2020.
.bib
@article{AlexanderssonUhlin2020, author = {Alexandersson, Per and Uhlin, Joakim}, title = {Cyclic sieving, skew {M}acdonald polynomials and {S}chur positivity}, journal = {Algebraic Combinatorics}, publisher = {MathOA foundation}, volume = {3}, number = {4}, year = {2020}, pages = {913-939}, doi = {10.5802/alco.123}, language = {en}, url2 = {alco.centre-mersenne.org/item/ALCO_2020__3_4_913_0/} } - [And24]Portia Anderson. Commutative Properties of Schubert Puzzles with Convex Polygonal Boundary Shapes. arXiv:2404.06320, 2024.
.bib
@article{Anderson2024Puzzlesx, author = {Portia Anderson}, title = {Commutative {P}roperties of {S}chubert {P}uzzles with {C}onvex {P}olygonal {B}oundary {S}hapes}, year = {2024}, eprint = {2404.06320}, url = {https://arxiv.org/abs/2404.06320}, journal = {arXiv e-prints} } - [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.
.bib
@article{Azenhas1999, doi = {10.1080/03081089908818605}, url2 = {https://doi.org/10.1080/03081089908818605}, year = {1999}, month = jul, publisher = {Informa {UK} Limited}, volume = {46}, number = {1-2}, pages = {51--99}, author = {Olga Azenhas}, title = {The admissible interval for the invariant factors of a product of matrices}, journal = {Linear and Multilinear Algebra} } - [ACM25]Olga Azenhas, Alessandro Conflitti and Ricardo Mamede. A uniform action of the dihedral group ${Z}_2\times {D}_3$ on Littlewood–Richardson coefficients. arXiv:2501.01947, 2025.
.bib
@article{AzenhasConflittiMamede2025x, author = {Olga Azenhas and Alessandro Conflitti and Ricardo Mamede}, title = {A uniform action of the dihedral group ${Z}_2\times {D}_3$ on {L}ittlewood--{R}ichardson coefficients}, year = {2025}, eprint = {2501.01947}, url = {https://arxiv.org/abs/2501.01947}, journal = {arXiv e-prints} } - [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.
.bib
@article{BerensteinZelevinsky1988, title = {Tensor product multiplicities and convex polytopes in partition space}, journal = {Journal of Geometry and Physics}, volume = {5}, number = {3}, pages = {453--472}, year = {1988}, issn = {0393-0440}, doi = {10.1016/0393-0440(88)90033-2}, url2 = {http://www.sciencedirect.com/science/article/pii/0393044088900332}, author = {A. D. Berenstein and A. V. Zelevinsky} } - [BZ92]A. D. Berenstein and A. V. Zelevinsky. Triple multiplicities for $sl(r + 1)$ and the spectrum of the exterior algebra of the adjoint representation. Journal of Algebraic Combinatorics, 1(1):7–22, 1992.
.bib
@article{BerensteinZelevinsky1992, year={1992}, issn={0925-9899}, journal={Journal of Algebraic Combinatorics}, volume={1}, number={1}, doi={10.1023/A:1022429213282}, title={Triple Multiplicities for $sl(r + 1)$ and the Spectrum of the Exterior Algebra of the Adjoint Representation}, publisher={Kluwer Academic Publishers-Plenum Publishers}, author={Berenstein, A. D. and Zelevinsky, A. V.}, pages={7--22}, language={English} } - [BR20]Emmanuel Briand and Mercedes Rosas. The 144 symmetries of the Littlewood-Richardson coefficients of ${{S}{L}}_3$. arXiv:2004.04995, 2020.
.bib
@article{BriandRosas2020x, author = {Emmanuel Briand and Mercedes Rosas}, title = {The 144 symmetries of the {L}ittlewood-{R}ichardson coefficients of ${{S}{L}}_3$}, year = {2020}, eprint = {2004.04995}, url = {https://arxiv.org/abs/2004.04995}, journal = {arXiv e-prints} } - [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.
.bib
@article{Buch2000, author = {Anders Skovsted Buch}, title = {The saturation conjecture (after {A}. {K}nutson and {T}. {T}ao), with an appendix by {W}illiam {F}ulton}, journal = {Enseign. Math. (2)}, volume = {46}, pages={43--60}, year = {2000} } - [Buc]Anders Skovsted Buch. lrcalc: Littlewood–richardson calculator. . Software
.bib
@misc{BuchLRCalc, author = {Anders Skovsted Buch}, title = {\texttt{lrcalc}: Littlewood--Richardson Calculator}, url = {https://sites.math.rutgers.edu/~asbuch/lrcalc/}, note = {Software} } - [BI13]Peter Bürgisser and Christian Ikenmeyer. Deciding positivity of Littlewood–Richardson coefficients. SIAM Journal on Discrete Mathematics, 27(4):1639–1681, January 2013.
.bib
@article{BurgisserIkenmeyer2013, doi = {10.1137/120892532}, url2 = {https://doi.org/10.1137/120892532}, year = {2013}, month = jan, publisher = {Society for Industrial \& Applied Mathematics ({SIAM})}, volume = {27}, number = {4}, pages = {1639--1681}, author = {Peter B{\"{u}}rgisser and Christian Ikenmeyer}, title = {Deciding Positivity of {L}ittlewood--{R}ichardson Coefficients}, journal = {{SIAM} Journal on Discrete Mathematics} } - [CR25]Pierre-Emmanuel Chaput and Nicolas Ressayre. Bidilatation of small Littlewood–Richardson coefficients. Representation Theory, 29(12):379–393, July 2025.
.bib
@article{ChaputRessayre2025, title={Bidilatation of Small {L}ittlewood--{R}ichardson Coefficients}, volume={29}, ISSN={1088-4165}, DOI={10.1090/ert/691}, number={12}, journal={Representation Theory}, publisher={American Mathematical Society (AMS)}, author={Chaput, Pierre-Emmanuel and Ressayre, Nicolas}, year={2025}, month=jul, pages={379--393}, eprint = {2206.03054} } - [CZ11]Robert Coquereaux and Jean-Bernard Zuber. On sums of tensor and fusion multiplicities. Journal of Physics A: Mathematical and Theoretical, 44(29):295208, June 2011.
.bib
@article{CoquereauxZuber2011, doi = {10.1088/1751-8113/44/29/295208}, url2 = {https://doi.org/10.1088%2F1751-8113%2F44%2F29%2F295208}, year = 2011, month = {jun}, publisher = {{IOP} Publishing}, volume = {44}, number = {29}, pages = {295208}, author = {Robert Coquereaux and Jean-Bernard Zuber}, title = {On sums of tensor and fusion multiplicities}, journal = {Journal of Physics A: Mathematical and Theoretical} } - [DW02]Harm Derksen and Jerzy Weyman. On the Littlewood–Richardson polynomials. Journal of Algebra, 255(2):247–257, 2002.
.bib
@article{DerksenWeyman2002, title = {On the {L}ittlewood--{R}ichardson polynomials}, journal = {Journal of Algebra }, volume = {255}, number = {2}, pages = {247--257}, year = {2002}, issn = {0021-8693}, doi = {10.1016/S0021-8693(02)00125-4}, url2 = {http://www.sciencedirect.com/science/article/pii/S0021869302001254}, author = {Harm Derksen and Jerzy Weyman} } - [FG93]Sergey Fomin and Curtis Greene. A Littlewood–Richardson miscellany. European Journal of Combinatorics, 14(3):191–212, May 1993.
.bib
@article{FominGreene1993, doi = {10.1006/eujc.1993.1024}, url2 = {https://doi.org/10.1006/eujc.1993.1024}, year = {1993}, month = may, publisher = {Elsevier {BV}}, volume = {14}, number = {3}, pages = {191--212}, author = {Sergey Fomin and Curtis Greene}, title = {A {L}ittlewood--{R}ichardson Miscellany}, journal = {European Journal of Combinatorics} } - [Ful97]William Fulton. Young tableaux: With applications to representation theory and geometry. London mathematical society student texts (book 35). Cambridge University Press, 1997.
.bib
@book{Fulton1997, title = {Young Tableaux: With Applications to Representation Theory and Geometry}, doi = {10.1017/cbo9780511626241}, author = {William Fulton}, isbn = {978-0511626241}, series = {London Mathematical Society Student Texts (Book 35)}, year = {1997}, publisher = {Cambridge University Press} } - [GG25]Christian Gaetz and Yibo Gao. Interlacing Triangles, Schubert Puzzles, and Graph Colorings. Communications in Mathematical Physics, 406(5), 2025.
.bib
@article{GaetzGao2025, author = {Gaetz, Christian and Gao, Yibo}, title = {Interlacing {T}riangles, {S}chubert {P}uzzles, and {G}raph {C}olorings}, year = {2025}, journal = {Communications in Mathematical Physics}, volume = {406}, number = {5}, publisher = {Springer Science and Business Media LLC}, doi = {10.1007/s00220-025-05312-7}, url = {http://dx.doi.org/10.1007/s00220-025-05312-7}, issn = {1432-0916}, eprint = {2408.07863} } - [GOY20]Shiliang Gao, Gidon Orelowitz and Alexander Yong. Newell–Littlewood numbers II: Extended Horn inequalities. arXiv:2009.09904, 2020.
.bib
@article{GaoOrelowitzYong2020IIx, Author = {Shiliang Gao and Gidon Orelowitz and Alexander Yong}, Title = {Newell--{L}ittlewood numbers {II}: extended {H}orn inequalities}, Year = {2020}, Eprint = {2009.09904}, url = {https://arxiv.org/abs/2009.09904}, journal = {arXiv e-prints} } - [GOY20]Shiliang Gao, Gidon Orelowitz and Alexander Yong. Newell–Littlewood numbers. arXiv:2005.09012, 2020.
.bib
@article{GaoOrelowitzYong2020x, Author = {Shiliang Gao and Gidon Orelowitz and Alexander Yong}, Title = {Newell--{L}ittlewood numbers}, Year = {2020}, Eprint = {2005.09012}, url = {https://arxiv.org/abs/2005.09012}, journal = {arXiv e-prints} } - [Gri21]Darij Grinberg. The Pelletier–Ressayre hidden symmetry for Littlewood–Richardson coefficients. Combinatorial Theory, 1(0), 2021.
.bib
@article{Grinberg2021, author = {Darij Grinberg}, title = {The {P}elletier--{R}essayre hidden symmetry for {L}ittlewood--{R}ichardson coefficients}, year = {2021}, journal = {Combinatorial Theory}, volume = {1}, number = {0}, publisher = {California Digital Library (CDL)}, doi = {10.5070/c61055382}, url = {https://doi.org/10.5070/C61055382} } - [HW14]Pamela E. Harris and Jeb F. Willenbring. Sums of squares of Littlewood–Richardson coefficients and $\mathrm{GL}_n$-harmonic polynomials. Symmetry: Representation theory and its applications, 257:305–326, 2014.
.bib
@incollection{HarrisWillenbring2014, author = {Pamela E. Harris and Jeb F. Willenbring}, title = {Sums of squares of {Littlewood--Richardson} coefficients and {$\mathrm{GL}_n$}-harmonic polynomials}, booktitle = {Symmetry: Representation Theory and Its Applications}, series = {Progress in Mathematics}, volume = {257}, pages = {305--326}, year = {2014}, doi = {10.1007/978-1-4939-1590-3_11} } - [Ike16]Christian Ikenmeyer. Small Littlewood–Richardson coefficients. Journal of Algebraic Combinatorics, 44(1):1–29, February 2016.
.bib
@article{Ikenmeyer2016, doi = {10.1007/s10801-015-0658-2}, url2 = {https://doi.org/10.1007/s10801-015-0658-2}, year = {2016}, month = feb, publisher = {Springer Science and Business Media {LLC}}, volume = {44}, number = {1}, pages = {1--29}, author = {Christian Ikenmeyer}, title = {Small {L}ittlewood--{R}ichardson coefficients}, journal = {Journal of Algebraic Combinatorics} } - [KM19]Ryan Kaliszewski and Jennifer Morse. Colorful combinatorics and Macdonald polynomials. European Journal of Combinatorics, 81:354–377, October 2019.
.bib
@article{KaliszewskiMorse2019, doi = {10.1016/j.ejc.2019.05.006}, url2 = {https://doi.org/10.1016/j.ejc.2019.05.006}, year = {2019}, month = oct, publisher = {Elsevier {BV}}, volume = {81}, pages = {354--377}, author = {Ryan Kaliszewski and Jennifer Morse}, title = {Colorful combinatorics and {M}acdonald polynomials}, journal = {European Journal of Combinatorics} } - [Kin21]Ronald C King. Stretched Newell–Littlewood coefficients. arXiv:2101.00984, 2021.
.bib
@article{King2021x, Author = {Ronald C King}, Title = {Stretched {N}ewell--{L}ittlewood coefficients}, Year = {2021}, Eprint = {2101.00984}, url = {https://arxiv.org/abs/2101.00984}, journal = {arXiv e-prints} } - [KTT04]R. C. King, C. Tollu and F. Toumazet. Stretched Littlewood–Richardson coefficients and Kostka coefficients. Symmetry in physics: In memory of Robert T. Sharp, 34:99–112, 2004.
.bib
@incollection{KingTolluToumazet2004, author = {R. C. King and C. Tollu and F. Toumazet}, title = {Stretched {L}ittlewood--{R}ichardson coefficients and {K}ostka coefficients}, journal = {CRM Proceedings and Lecture Notes}, year = {2004}, pages = {99--112}, volume = {34}, booktitle = {Symmetry in Physics: In Memory of {R}obert {T}. {S}harp}, isbn = {978-0-8218-3409-1}, editor = {P. Winternitz and J. Harnard and C. S. Lam and J. Patera}, publisher = {AMS / OUP}, url = {http://eprints.soton.ac.uk/41202/} } - [Kly98]A. A. Klyachko. Stable bundles, representation theory and Hermitian operators. Selecta Mathematica, 4(3):419–445, September 1998.
.bib
@article{Klyachko1998, doi = {10.1007/s000290050037}, url2 = {https://doi.org/10.1007/s000290050037}, year = {1998}, month = sep, publisher = {Springer Science and Business Media {LLC}}, volume = {4}, number = {3}, pages = {419--445}, author = {A. A. Klyachko}, title = {Stable bundles, representation theory and {H}ermitian operators}, journal = {Selecta Mathematica} } - [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.
.bib
@article{KnutsonTaoWoodward2004, doi = {10.1090/s0894-0347-03-00441-7}, url2 = {https://doi.org/10.1090/s0894-0347-03-00441-7}, year = {2004}, month = jan, publisher = {American Mathematical Society ({AMS})}, volume = {17}, number = {1}, pages = {19--49}, title = {The honeycomb model of ${GL}_n({C})$ tensor products {II}: Puzzles determine facets of the {L}ittlewood--{R}ichardson cone}, author = {Allen Knutson and Terence Tao and Christopher Woodward}, journal = {Journal of the American Mathematical Society} } - [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.
.bib
@article{KnutsonTau1999, doi = {10.1090/s0894-0347-99-00299-4}, url2 = {https://doi.org/10.1090/s0894-0347-99-00299-4}, year = {1999}, month = oct, publisher = {American Mathematical Society ({AMS})}, volume = {12}, number = {04}, pages = {1055--1091}, author = {Allen Knutson and Terence Tao}, title = {The honeycomb model of ${GL}_n({C})$ tensor products {I}: proof of the saturation conjecture}, journal = {Journal of the American Mathematical Society} } - [KS22]Justyna Kosakowska and Markus Schmidmeier. The socle tableau as a dual version of the Littlewood–Richardson tableau. Journal of the London Mathematical Society, May 2022.
.bib
@article{KosakowskaSchmidmeier2022, doi = {10.1112/jlms.12601}, url2 = {https://doi.org/10.1112%2Fjlms.12601}, year = 2022, month = {may}, publisher = {Wiley}, author = {Justyna Kosakowska and Markus Schmidmeier}, title = {The socle tableau as a dual version of the {L}ittlewood--{R}ichardson tableau}, journal = {Journal of the London Mathematical Society} } - [KRV21]Mrigendra Singh Kushwaha, K. N. Raghavan and Sankaran Viswanath. The saturation problem for refined Littlewood–Richardson coefficients. Séminaire Lotharingien de Combinatoire, 85B:Article 52, 12 pp., 2021.
.bib
@article{KushwahaRaghavanViswanath2021, author = {Mrigendra Singh Kushwaha and K. N. Raghavan and Sankaran Viswanath}, title = {The saturation problem for refined {L}ittlewood--{R}ichardson coefficients}, year = {2021}, journal = {Séminaire Lotharingien de Combinatoire}, volume = {85B}, pages = {Article 52, 12 pp.}, url = {https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2021/52Kushwaha.pdf} } - [LN26]Tuong Le and Son Nguyen. Skew Hives, Skew Skeps, Skew Schur Log-Concavity. arXiv:2608.13544v1, 2026.
.bib
@article{LeNguyen2026x, author = {Tuong Le and Son Nguyen}, title = {Skew {H}ives, {S}kew {S}keps, {S}kew {S}chur {L}og-{C}oncavity}, year = {2026}, eprint = {2608.13544v1}, url = {https://arxiv.org/abs/2608.13544v1}, journal = {arXiv e-prints} } - [Lot02]M. Lothaire. Algebraic combinatorics on words. Cambridge University Press, 2002.
.bib
@book{Lothaire2002, doi = {10.1017/cbo9781107326019}, year = {2002}, publisher = {Cambridge University Press}, author = {M. Lothaire}, title = {Algebraic Combinatorics on Words} } - [Min24]Jaewon Min. Proof of the Newell-Littlewood saturation conjecture. arXiv:2409.00233, 2024.
.bib
@article{Min2024x, Author = {Jaewon Min}, Title = {Proof of the {N}ewell-{L}ittlewood saturation conjecture}, Year = {2024}, Eprint = {2409.00233}, url = {https://arxiv.org/abs/2409.00233}, journal = {arXiv e-prints} } - [MS99]Alexander I. Molev and Bruce E. Sagan. A Littlewood–Richardson rule for factorial Schur functions. Transactions of the American Mathematical Society, 351(11):4429–4444, November 1999.
.bib
@article{MolevSagan1999, doi = {10.1090/s0002-9947-99-02381-8}, url2 = {https://doi.org/10.1090/s0002-9947-99-02381-8}, year = {1999}, month = nov, title = {A {L}ittlewood--{R}ichardson Rule for factorial {S}chur functions}, publisher = {American Mathematical Society ({AMS})}, volume = {351}, number = {11}, pages = {4429--4444}, author = {Alexander I. Molev and Bruce E. Sagan}, journal = {Transactions of the American Mathematical Society} } - [Nar06]Hariharan Narayanan. On the complexity of computing Kostka numbers and Littlewood–Richardson coefficients. Journal of Algebraic Combinatorics, 24(3):347–354, July 2006.
.bib
@article{Narayanan2006, doi = {10.1007/s10801-006-0008-5}, url2 = {https://doi.org/10.1007/s10801-006-0008-5}, year = {2006}, month = jul, publisher = {Springer Science and Business Media {LLC}}, volume = {24}, number = {3}, pages = {347--354}, author = {Hariharan Narayanan}, title = {On the complexity of computing {K}ostka numbers and {L}ittlewood--{R}ichardson coefficients}, journal = {Journal of Algebraic Combinatorics} } - [NNW25]Chau Nguyen, Son Nguyen and Dora Woodruff. Shuffle tableaux, Littlewood–Richardson coefficients, and Schur log-concavity. arXiv:2506.00349, 2025.
.bib
@article{NguyenNguyenWoodruff2025x, author = {Chau Nguyen and Son Nguyen and Dora Woodruff}, title = {Shuffle Tableaux, {L}ittlewood--{R}ichardson Coefficients, and {S}chur Log-Concavity}, year = {2025}, eprint = {2506.00349}, url = {https://arxiv.org/abs/2506.00349}, journal = {arXiv e-prints} } - [PS26]Igor Pak and Daniel Soskin. On the largest Littlewood–Richardson coefficient. arXiv:2608.03281v1, 2026.
.bib
@article{PakSoskin2026x, author = {Igor Pak and Daniel Soskin}, title = {On the largest {L}ittlewood--{R}ichardson coefficient}, year = {2026}, eprint = {2608.03281v1}, url = {https://arxiv.org/abs/2608.03281v1}, journal = {arXiv e-prints} } - [PV05]Igor Pak and Ernesto Vallejo. Combinatorics and geometry of Littlewood–Richardson cones. European Journal of Combinatorics, 26(6):995–1008, 2005.
.bib
@article{PakVallejo2005, author = {Igor Pak and Ernesto Vallejo}, title = {Combinatorics and geometry of {L}ittlewood--{R}ichardson cones}, year = {2005}, journal = {European Journal of Combinatorics}, volume = {26}, number = {6}, pages = {995--1008}, doi = {10.1016/j.ejc.2004.06.008}, url = {https://doi.org/10.1016/j.ejc.2004.06.008} } - [PR22]Maxime Pelletier and Nicolas Ressayre. Some Unexpected Properties of Littlewood-Richardson Coefficients. The Electronic Journal of Combinatorics, 29(4), 2022.
.bib
@article{PelletierRessayre2022, author = {Maxime Pelletier and Nicolas Ressayre}, title = {Some {U}nexpected {P}roperties of {L}ittlewood-{R}ichardson {C}oefficients}, year = {2022}, journal = {The Electronic Journal of Combinatorics}, volume = {29}, number = {4}, publisher = {The Electronic Journal of Combinatorics}, doi = {10.37236/9928}, url = {https://doi.org/10.37236/9928} } - [Ras04]Etienne Rassart. A polynomiality property for Littlewood–Richardson coefficients. J. Comb. Theory Ser. A, 107(2):161–179, August 2004.
.bib
@article{Rassart2004, author = {Etienne Rassart}, title = {A polynomiality property for {L}ittlewood--{R}ichardson coefficients}, journal = {J. Comb. Theory Ser. A}, issue_date = {August 2004}, volume = {107}, number = {2}, month = aug, year = {2004}, issn = {0097-3165}, pages = {161--179}, numpages = {19}, doi = {10.1016/j.jcta.2004.04.003}, acmid = {1032085}, publisher = {Academic Press, Inc.}, address = {Orlando, FL, USA}, } - [Sam]Matthew J. Samuel. schubmult: Compute products of schubert polynomials. . Python package
.bib
@misc{SamuelSchubmult, author = {Matthew J. Samuel}, title = {\texttt{schubmult}: Compute Products of Schubert Polynomials}, url = {https://pypi.org/project/schubmult/}, note = {Python package} } - [She15]Cass Sherman. Geometric proof of a conjecture of King, Tollu, and Toumazet. arXiv:1505.06551, 2015.
.bib
@article{Sherman2015x, Author = {Cass Sherman}, Title = {Geometric Proof of a Conjecture of {K}ing, {T}ollu, and {T}oumazet}, Year = {2015}, Eprint = {1505.06551}, url = {https://arxiv.org/abs/1505.06551}, journal = {arXiv e-prints} } - [She17]Cass Sherman. Quiver generalization of a conjecture of King, Tollu, and Toumazet. Journal of Algebra, 480:487–504, June 2017.
.bib
@article{Sherman2017, doi = {10.1016/j.jalgebra.2017.03.011}, url2 = {https://doi.org/10.1016/j.jalgebra.2017.03.011}, year = {2017}, month = jun, publisher = {Elsevier {BV}}, volume = {480}, pages = {487--504}, author = {Cass Sherman}, title = {Quiver generalization of a conjecture of {K}ing, {T}ollu, and {T}oumazet}, journal = {Journal of Algebra} } - [Shr22]Sagar Shrivastava. Littlewood–Richardson coefficients as a signed sum of Kostka numbers. arXiv:2211.10669, 2022.
.bib
@article{Shrivastava2022x, Author = {Sagar Shrivastava}, Title = {Littlewood--{R}ichardson coefficients as a signed sum of {K}ostka numbers}, Year = {2022}, journal = {arXiv e-prints}, Eprint = {2211.10669} } - [Spe26]David E. Speyer. L-log-concavity and a proof of the conjecture of Lam, Postnikov and Pylyavskyy. arXiv:2601.05007, 2026.
.bib
@article{Speyer2026x, author = {David E. Speyer}, title = {L-log-concavity and a proof of the conjecture of {L}am, {P}ostnikov and {P}ylyavskyy}, year = {2026}, eprint = {2601.05007}, url = {https://arxiv.org/abs/2601.05007}, journal = {arXiv e-prints} } - [Sta01]Richard P. Stanley. Enumerative Combinatorics: Volume 2. Cambridge University Press, First, 2001.
.bib
@book{StanleyEC2, author = {Richard P. Stanley}, title = {Enumerative {C}ombinatorics: {V}olume 2}, publisher = {Cambridge University Press}, year = {2001}, edition = {First}, isbn = {0521789877}, doi = {10.1017/CBO9780511609589} } - [Ste61]Robert Steinberg. A general Clebsch–Gordan theorem. Bull. Amer. Math. Soc., 67(4):406–407, July 1961.
.bib
@article{Steinberg1961, author = {Robert Steinberg}, fjournal = {Bulletin of the American Mathematical Society}, journal = {Bull. Amer. Math. Soc.}, month = jul, number = {4}, pages = {406--407}, publisher = {American Mathematical Society}, title = {A general {C}lebsch--{G}ordan theorem}, url = {https://projecteuclid.org:443/euclid.bams/1183524263}, volume = {67}, year = {1961} } - [TG22]Müge Taşkın and R. Bedii Gümüş. Lower bounds for nonzero Littlewood–Richardson coefficients. arXiv:2208.06541, 2022.
.bib
@article{TaskinGumus2022x, Author = {M{\"{u}}ge Taşk{\i}n and R. Bedii G{\"{u}}m{\"{u}}ş}, Title = {Lower bounds for Nonzero {L}ittlewood--{R}ichardson coefficients}, Year = {2022}, Eprint = {2208.06541}, url = {https://arxiv.org/abs/2208.06541}, journal = {arXiv e-prints} } - [Tha22]Warut Thawinrak. Polynomiality of the stretched Littlewood–Richardson coefficients. arXiv:2211.06810, 2022.
.bib
@article{Thawinrak2022x, Author = {Warut Thawinrak}, Title = {Polynomiality of the Stretched {L}ittlewood--{R}ichardson Coefficients}, Year = {2022}, Eprint = {2211.06810}, url = {https://arxiv.org/abs/2211.06810}, journal = {arXiv e-prints} } - [Wen21]Xueqing Wen. A closed formula of Littlewood–Richardson coefficients. arXiv:2107.09907, 2021.
.bib
@article{Wen2021x, Author = {Xueqing Wen}, Title = {A closed formula of {L}ittlewood--{R}ichardson coefficients}, Year = {2021}, Eprint = {2107.09907}, url = {https://arxiv.org/abs/2107.09907}, journal = {arXiv e-prints} }