#Catalan symmetric functions
Catalan symmetric functions were introduced in [Che10] and [Pan10]. These functions are \(GL_\ell\)-equivariant Euler characteristics of vector bundles on the flag variety. The Catalan symmetric functions specialize to \(k\)-Schur functions, see [BMPS18].
See also J. Blasiak’s FPSAC 2023 talk [Bla23].
#Definition
Let \(R_{ij}\) be the raising operators on Schur functions, so that \[R_{ij} \schurS_{\alpha} = \schurS_{\alpha+\varepsilon_i-\varepsilon_j}.\] Here, \(\alpha\) can be a composition, and we then evaluate the Schur function using the Jacobi–Trudi formula.
A root ideal \(\Phi\) is an upper order ideal in \[\Delta_\ell^+ \coloneqq \{ (i,j) : 1 \leq i \lt j \leq \ell \}\] with the partial order relation \((a,b) \leq (c,d)\) when \(a\geq c\) and \(b \leq d.\) There are \(\catalan(\ell)\) different such order ideals, thus explaining the name.
Example
For example, \(\{ 15, 25, 35, 14, 24 \}\) is an order ideal: it is the set of entries above a Dyck path in the diagram
See also how area sequences of length \(\ell\) are related to unit interval graphs.
Let \(\gamma \in \setZ^\ell\) and let \(\Phi\) be a root ideal. The Catalan symmetric function is then defined as \[\catalanH_{\Phi,\gamma}(\xvec;t) \coloneqq \prod_{(i,j)\in \Phi} (1-t R_{ij})^{-1}\schurS_{\gamma}(\xvec).\] Note that \[(1-tR_{ij})^{-1} = 1 + tR_{ij} + t^2 (R_{ij})^2 + t^3 (R_{ij})^3 + \dotsb\] but one only has to apply a finite number of these as \((R_{ij})^k\) kills any Schur function for sufficiently large \(k.\) By construction, \(\catalanH_{\Phi,\gamma}(\xvec;t)\) is a symmetric function.
The Catalan symmetric functions generalize the transformed Hall–Littlewood polynomials. If \(\mu\) is a partition of \(n,\) then using the full set of roots gives \[\hallLittlewoodT_{\mu}(\xvec;q) = \catalanH_{\Delta_n^+,\mu}(\xvec;q).\]
Example
For example, \(\Phi = \{ 15, 25, 35, 14, 24 \}\) and \(\gamma=(4,2,1,1,0)\) gives \[\catalanH_{\Phi,\gamma}(\xvec;t) = \schurS_{4211} + t \schurS_{431} + t \schurS_{521}\]
In [Che10], it was conjectured that \(\catalanH_{\Phi,\mu}(\xvec;t)\) is Schur-positive for any \(\Phi\) and partition \(\mu.\) This conjecture is resolved by J. Blasiak, J. Morse, and A. Pun in [BMP20]. They introduce a larger family of non-symmetric Catalan functions, the tame non-symmetric Catalan functions, \(\catalanH_{\Phi,\mu,w}(\xvec;t),\) which depend on an additional parameter \(w \in \symS_n.\) It is then proved that \(\catalanH_{\Phi,\mu,w}(\xvec;t)\) is key-positive, which then implies the Schur positivity for \(\catalanH_{\Phi,\mu}(\xvec;t).\)
#Katalan symmetric functions
In [BMS20], the authors consider a \(K\)-theoretic version of the Catalan symmetric functions, named Katalan symmetric functions, and show that this family includes the \(K\)-\(k\)-Schur functions and the usual Catalan symmetric functions.
The definition again uses raising operators, together with lowering operators. Let \(\Phi \subseteq \Delta_\ell^+\) be a root ideal, let \(M\) be a multiset on \(\{1,\dotsc,\ell\},\) and let \(\gamma \in \setZ^\ell.\) If \(g_\gamma\) denotes the determinantally extended dual stable Grothendieck function, and \(L_j g_\gamma = g_{\gamma-\varepsilon_j},\) then the Katalan symmetric function is \[\catalanH_{\Phi,M,\gamma}(\xvec) \coloneqq \prod_{j\in M}(1-L_j) \prod_{(i,j)\in \Phi}(1-R_{ij})^{-1} g_\gamma(\xvec).\] Repeated entries of \(M\) contribute repeated lowering factors.
A conjecture regarding the Katalan symmetric functions is solved in [IIN24].
#\(k\)-Schur polynomials
The \(k\)-Schur functions were introduced in [LLM03] under a different name, using the notation \(A_\mu^{(k)}[\xvec;t].\) The motivation was to provide a strong refinement of the Schur-positivity conjecture of the modified Macdonald polynomials. Several alternative definitions of \(k\)-Schur functions have since surfaced, and not all have been proved to be equivalent. For a thorough reference, see the book [LLMS+14].
For each integer \(k,\) we have the family of \(k\)-Schur functions \(\{ \kSchur^{(k)}_\lambda(\xvec) \}\) where the \(\lambda\) are \(k\)-bounded partitions, meaning that \(\lambda_1 \leq k.\) The \(k\)-Schur functions form a basis in the subring of symmetric functions, spanned by \(\completeH_1,\dotsc,\completeH_k,\) the complete homogeneous symmetric functions.
Y. Fang and X. Gao prove cases of the alternating dual Pieri rule and \(k\)-branching conjectures for closed \(k\)-Schur Katalan functions [FG25]. In particular, they prove the branching conjecture for strictly decreasing partitions. Y. Fan and X. Gao introduce weighted \(K\)-\(k\)-Schur functions inside the class of Katalan functions [FG25]. These simultaneously extend \(K\)-\(k\)-Schur functions and closed \(k\)-Schur Katalan functions, and give new cases of the \(K\)-\(k\)-Schur alternating conjecture.
In [LM07], it is shown that whenever the hook-length of \(\lambda\) is no larger than \(k,\) we have the identity \[\kSchur^{(k)}_\lambda(\xvec) = \schurS_\lambda(\xvec).\] Hence, as \(k\to \infty,\) the \(k\)-Schur functions reduce to the usual Schur functions.
#Definition
Note that there are several different, but conjecturally equivalent, definitions of \(k\)-Schur functions. We use the definition in [BMPS18], which defines the \(k\)-Schur functions as Catalan symmetric functions for special root ideals. Let \(\mu\) be a partition with at most \(\ell\) parts and \(\mu_1 \leq k.\)
Let \(\Phi_\mu \coloneqq \{ (i,j) \in \Delta_\ell^+ : k-\mu_i+i \lt j \}\) and define the \(k\)-Schur functions as \[\kSchur^{(k)}_\mu(\xvec;t) \coloneqq \catalanH_{\Phi_\mu,\mu}(\xvec;t) = \prod_{i=1}^\ell \prod_{j=k+1-\mu_i+i}^\ell (1-tR_{ij})^{-1} \schurS_\mu.\]
The specialization \(t=1\) is also called \(k\)-Schur functions, \[\kSchur^{(k)}_\mu(\xvec) \coloneqq \kSchur^{(k)}_\mu(\xvec;1).\] Most results so far concern this specialization.
Example
We have the following Schur expansions of some \(k\)-Schur functions: \[\kSchur^{(2)}_{221}(\xvec) = \schurS_{221}(\xvec)+\schurS_{311}(\xvec)+2\schurS_{320}(\xvec) +2\schurS_{410}(\xvec)+\schurS_{500}(\xvec)\] \[\kSchur^{(3)}_{221}(\xvec) = \schurS_{221}(\xvec)+\schurS_{320}(\xvec)\quad \text{ and } \quad \kSchur^{(4)}_{221}(\xvec) = \schurS_{221}(\xvec)\]
#Relation with affine Stanley symmetric functions
In [Lam06] it is shown that the \(k\)-Schur functions are dual to the affine Schur functions.
T. Ikeda, M. Shimozono, and K. Yamaguchi introduce \(K\)-theoretic double \(k\)-Schur functions as Schubert bases for the torus-equivariant \(K\)-homology of the affine Grassmannian [ISY24]. Their construction is a \(K\)-theoretic double analogue of the affine Grassmannian Schur-basis story.
M. Takigiku studies the sums \(\sum_{\mu\leq \lambda} g^{(k)}_\mu\) of \(K\)-\(k\)-Schur functions over principal order ideals in the strong Bruhat order on \(k\)-bounded partitions [Tak19]. He proves a Pieri-type formula for these sums and a \(k\)-rectangle factorization formula analogous to the usual \(k\)-Schur factorization.
#Pieri rule
L. Lapointe and J. Morse proved the following Pieri rule.
Theorem (See [Thm. 29, LM07]).
Let \(\nu\) be a \(k\)-bounded partition and \(r \leq k.\) Then \[\completeH_r \kSchur^{(k)}_\nu = \sum_{\mu \in H^k_{\nu,r}} \kSchur^{(k)}_\mu\] where \(H^k_{\nu,r}\) is a certain subset of partitions formed by adding horizontal \(r\)-strips to \(\lambda.\) More precisely, \[H^k_{\nu,r} = \{ \mu : \mu/\nu \text{ is a horizontal strip and } \mu^{(k)}/\nu^{(k)} \text{ is a vertical $r$-strip} \}.\] Here, \(\mu^{(k)}\) denotes the \(k\)-conjugate of \(\mu.\)
#Murnaghan–Nakayama rule
J. Bandlow, A. Schilling, and M. Zabrocki prove the following analog of the Murnaghan–Nakayama rule.
Theorem (See [BSZ11]).
For \(1\leq r \leq k\) and \(\lambda\) being a \(k\)-bounded partition, \[\powerSum_r \kSchur^{(k)}_\lambda = \sum_{\mu} (-1)^{\ht(\mu/\lambda)} \kSchur^{(k)}_\mu\] where the sum is over all \(k\)-bounded partitions \(\mu\) such that \(\mu/\lambda\) is a \(k\)-ribbon of size \(r.\) The definition of a \(k\)-ribbon is somewhat involved, see [BSZ11] for details.
An alternative proof is given by S. J. Lee [Thm. 10.1, Lee15], and D.-K. Nguyen [Ngu22] gives a proof in the more general setting, where a Murnaghan–Nakayama rule for \(K\text{-}k\)-Schur functions is also provided.
#Littlewood–Richardson rule
It is conjectured that the coefficients \(c^{\nu,k}_{\lambda\mu}\) in \[\kSchur^{(k)}_\lambda \kSchur^{(k)}_\mu = \sum_{\nu : \nu_1 \leq k} c^{\nu,k}_{\lambda\mu} \kSchur^{(k)}_\nu\] are all non-negative. These coefficients are 3-point Gromov–Witten invariants, see [LM08], and thus sometimes proved to be non-negative.
#Schur expansion
It was conjectured in [LM07] that the \(k\)-Schur functions are Schur-positive. A stronger statement is that the \(k\)-Schur functions expand positively into \((k+1)\)-Schur functions. This is now proved in [Thm. 2.6, BMPS19], where an explicit combinatorial expansion of \(\kSchur^{(k)}_\lambda(\xvec)\) into \(\{ \kSchur^{(k+1)}_\mu(\xvec) \}_{\mu}\) is given.
#Relation to modified Macdonald polynomials
S. Kato [Kat25] shows that Garcia–Haiman modules can be decomposed into certain modules whose characters are \(k\)-Schur functions. This implies that a modified Macdonald polynomial indexed by a \(k\)-bounded partition expands positively into \(k\)-Schur functions. That is, it can be expressed as a linear combination of \(\kSchur^{(k)}_\lambda\) with coefficients in \(\setN[q,t]\); see [Cor. 9.4, Kat25].
Bibliography
- [BSZ11]Jason Bandlow, Anne Schilling and Mike Zabrocki. The Murnaghan–Nakayama rule for k-Schur functions. Journal of Combinatorial Theory, Series A, 118(5):1588–1607, July 2011.
.bib
@article{BandlowSchillingZabrocki2011, doi = {10.1016/j.jcta.2011.01.009}, url2 = {https://doi.org/10.1016/j.jcta.2011.01.009}, year = {2011}, month = jul, publisher = {Elsevier {BV}}, volume = {118}, number = {5}, pages = {1588--1607}, author = {Jason Bandlow and Anne Schilling and Mike Zabrocki}, title = {The {M}urnaghan--{N}akayama rule for k-{S}chur functions}, journal = {Journal of Combinatorial Theory, Series A} } - [Bla23]Jonah Blasiak. Catalania. 2023. FPSAC 2023 plenary talk, video
.bib
@misc{Blasiak2023CatalaniaVideo, author = {Jonah Blasiak}, title = {Catalania}, year = {2023}, url = {https://www.youtube.com/watch?v=YYOUHcgRC1E}, note = {FPSAC 2023 plenary talk, video} } - [BMP20]Jonah Blasiak, Jennifer Morse and Anna Pun. Demazure crystals and the Schur positivity of Catalan functions. arXiv:2007.04952, 2020.
.bib
@article{BlasiakMorsePun2020x, Author = {Jonah Blasiak and Jennifer Morse and Anna Pun}, Title = {Demazure crystals and the {S}chur positivity of {C}atalan functions}, Year = {2020}, Eprint = {2007.04952}, url = {https://arxiv.org/abs/2007.04952}, journal = {arXiv e-prints} } - [BMPS18]Jonah Blasiak, Jennifer Morse, Anna Pun and Daniel Summers. $k$-Schur expansions of Catalan functions. arXiv:1811.02490, 2018.
.bib
@article{BlasiakMorsePunSummers2018, Author = {Jonah Blasiak and Jennifer Morse and Anna Pun and Daniel Summers}, Title = {$k$-{S}chur expansions of {C}atalan functions}, Year = {2018}, Eprint = {1811.02490}, url = {https://arxiv.org/abs/1811.02490}, journal = {arXiv e-prints} } - [BMS20]Jonah Blasiak, Jennifer Morse and George H. Seelinger. $K$-theoretic Catalan functions. arXiv:2010.01759, 2020.
.bib
@article{BlasiakMorseSeelinger2020, Author = {Jonah Blasiak and Jennifer Morse and George H. Seelinger}, Title = {$K$-theoretic {C}atalan functions}, Year = {2020}, Eprint = {2010.01759}, url = {https://arxiv.org/abs/2010.01759}, journal = {arXiv e-prints} } - [BMPS19]Jonah Blasiak, Jennifer Morse, Anna Pun and Daniel Summers. Catalan functions and $k$-Schur positivity. Journal of the American Mathematical Society, 32(4):921–963, August 2019.
.bib
@article{BlasiakMorseSummers2019, doi = {10.1090/jams/921}, url2 = {https://doi.org/10.1090%2Fjams%2F921}, year = 2019, month = {aug}, publisher = {American Mathematical Society ({AMS})}, volume = {32}, number = {4}, pages = {921--963}, author = {Jonah Blasiak and Jennifer Morse and Anna Pun and Daniel Summers}, title = {Catalan functions and $k$-{S}chur positivity}, journal = {Journal of the American Mathematical Society} } - [Che10]Li-Chung Chen. Skew-linked partitions and a representation-theoretic model for $k$-Schur functions. UC Berkeley, 2010.
.bib
@PHDTHESIS{ChenLiChungThesis, author = {Li-Chung Chen}, title = {Skew-Linked Partitions and a Representation-Theoretic Model for $k$-{S}chur Functions}, school = {UC Berkeley}, year = {2010}, url={https://escholarship.org/uc/item/6dc2n1tj} } - [FG25]Yaozhou Fan and Xing Gao. Weighted $K$-$k$-Schur functions and their application to the $K$-$k$-Schur alternating conjecture. arXiv:2507.23222, 2025.
.bib
@article{FanGao2025x, author = {Yaozhou Fan and Xing Gao}, title = {Weighted {$K$}-$k$-{S}chur functions and their application to the {$K$}-$k$-{S}chur alternating conjecture}, year = {2025}, eprint = {2507.23222}, url = {https://arxiv.org/abs/2507.23222}, journal = {arXiv e-prints} } - [FG25]Yaozhou Fang and Xing Gao. Alternating dual Pieri rule conjecture and $k$-branching conjecture of closed $k$-Schur Katalan functions. arXiv:2501.04200, 2025.
.bib
@article{FangGao2025x, author = {Yaozhou Fang and Xing Gao}, title = {Alternating dual {P}ieri rule conjecture and $k$-branching conjecture of closed $k$-{S}chur {K}atalan functions}, year = {2025}, eprint = {2501.04200}, url = {https://arxiv.org/abs/2501.04200}, journal = {arXiv e-prints} } - [IIN24]Takeshi Ikeda, Shinsuke Iwao and Satoshi Naito. Closed $k$-Schur Katalan functions as $K$-homology Schubert representatives of the affine Grassmannian. Transactions of the American Mathematical Society, Series B, 11(20):667–702, March 2024.
.bib
@article{IkedaIwaoNaito2024, title = {Closed $k$-{S}chur {K}atalan functions as $K$-homology {S}chubert representatives of the affine {G}rassmannian}, volume = {11}, ISSN = {2330-0000}, url = {http://dx.doi.org/10.1090/btran/184}, DOI = {10.1090/btran/184}, number = {20}, journal = {Transactions of the American Mathematical Society, Series B}, publisher = {American Mathematical Society (AMS)}, author = {Ikeda, Takeshi and Iwao, Shinsuke and Naito, Satoshi}, year = {2024}, month = mar, pages = {667–702} } - [ISY24]Takeshi Ikeda, Mark Shimozono and Kohei Yamaguchi. Equivariant ${K}$-homology of affine Grassmannian and ${K}$-theoretic double $k$-Schur functions. arXiv:2408.10956, 2024.
.bib
@article{IkedaShimozonoYamaguchi2024x, author = {Takeshi Ikeda and Mark Shimozono and Kohei Yamaguchi}, title = {Equivariant ${K}$-homology of affine {G}rassmannian and ${K}$-theoretic double $k$-{S}chur functions}, year = {2024}, eprint = {2408.10956}, url = {https://arxiv.org/abs/2408.10956}, journal = {arXiv e-prints} } - [Kat25]Syu Kato. Categorification of $k$-Schur functions and refined Macdonald positivity. arXiv:2505.23202, 2025.
.bib
@article{Kato2025x, Author = {Syu Kato}, Title = {Categorification of $k$-{S}chur functions and refined {M}acdonald positivity}, Year = {2025}, Eprint = {2505.23202}, url = {https://arxiv.org/abs/2505.23202}, journal = {arXiv e-prints} } - [LLMS+14]Thomas Lam, Luc Lapointe, Jennifer Morse, Anne Schilling, Mark Shimozono and Mike Zabrocki. K-Schur functions and affine Schubert calculus (fields institute monographs). Springer, 2014.
.bib
@book{LLMSSZ2014, Author = {Thomas Lam and Luc Lapointe and Jennifer Morse and Anne Schilling and Mark Shimozono and Mike Zabrocki}, Title = {k-{S}chur Functions and Affine {S}chubert Calculus (Fields Institute Monographs)}, Publisher = {Springer}, Year = {2014}, ISBN = {978-1-4939-0682-6} } - [Lam06]Thomas Lam. Affine Stanley symmetric functions. American Journal of Mathematics, 128(6):1553–1586, 2006.
.bib
@article{Lam2006, doi = {10.1353/ajm.2006.0045}, url2 = {https://doi.org/10.1353/ajm.2006.0045}, year = {2006}, publisher = {Johns Hopkins University Press}, volume = {128}, number = {6}, pages = {1553--1586}, author = {Thomas Lam}, title = {Affine {S}tanley symmetric functions}, journal = {American Journal of Mathematics} } - [LLM03]Luc Lapointe, Alain Lascoux and Jennifer Morse. Tableau atoms and a new Macdonald positivity conjecture. Duke Mathematical Journal, 116(1):103–146, January 2003.
.bib
@article{LapointeLascouxMorse2003, doi = {10.1215/s0012-7094-03-11614-2}, url2 = {https://doi.org/10.1215/s0012-7094-03-11614-2}, year = {2003}, month = jan, publisher = {Duke University Press}, volume = {116}, number = {1}, pages = {103--146}, author = {Luc Lapointe and Alain Lascoux and Jennifer Morse}, title = {Tableau atoms and a new {M}acdonald positivity conjecture}, journal = {Duke Mathematical Journal} } - [LM07]Luc Lapointe and Jennifer Morse. A k-tableau characterization of k-Schur functions. Advances in Mathematics, 213(1):183–204, August 2007.
.bib
@article{LapointeMorse2007, doi = {10.1016/j.aim.2006.12.005}, url2 = {https://doi.org/10.1016/j.aim.2006.12.005}, year = {2007}, month = aug, publisher = {Elsevier {BV}}, volume = {213}, number = {1}, pages = {183--204}, author = {Luc Lapointe and Jennifer Morse}, title = {A k-tableau characterization of k-{S}chur functions}, journal = {Advances in Mathematics} } - [LM08]Luc Lapointe and Jennifer Morse. Quantum cohomology and the $k$-Schur basis. Transactions of the American Mathematical Society, 360(4):2021–2040, 2008.
.bib
@article{LapointeMorse2008, ISSN = {00029947}, URL = {http://www.jstor.org/stable/20161956}, author = {Luc Lapointe and Jennifer Morse}, journal = {Transactions of the American Mathematical Society}, number = {4}, pages = {2021--2040}, publisher = {American Mathematical Society}, title = {Quantum Cohomology and the {$k$}-{S}chur Basis}, volume = {360}, year = {2008} } - [Lee15]Seung Jin Lee. Combinatorial description of the cohomology of the affine flag variety. arXiv:1506.02390, 2015.
.bib
@article{Lee2015, Author = {Seung Jin Lee}, Title = {Combinatorial description of the cohomology of the affine flag variety}, Year = {2015}, Eprint = {1506.02390}, url = {https://arxiv.org/abs/1506.02390}, journal = {arXiv e-prints} } - [Ngu22]Duc-Khanh Nguyen. A generalization of the Murnaghan–Nakayama rule for $K$-$k$-Schur and $k$-Schur functions. arXiv:2212.02037, 2022.
.bib
@article{Nguyen2022x, Author = {Duc-Khanh Nguyen}, Title = {A generalization of the {M}urnaghan--{N}akayama rule for $K$-$k$-{S}chur and $k$-{S}chur functions}, Year = {2022}, Eprint = {2212.02037}, url = {https://arxiv.org/abs/2212.02037}, journal = {arXiv e-prints} } - [Pan10]Dmitri I. Panyushev. Generalised Kostka–Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles. Selecta Mathematica, 16(2):315–342, April 2010.
.bib
@article{Panyushev2010, doi = {10.1007/s00029-010-0022-2}, url2 = {https://doi.org/10.1007/s00029-010-0022-2}, year = {2010}, month = apr, publisher = {Springer Nature}, volume = {16}, number = {2}, pages = {315--342}, author = {Dmitri I. Panyushev}, title = {Generalised {K}ostka--{F}oulkes polynomials and cohomology of line bundles on homogeneous vector bundles}, journal = {Selecta Mathematica} } - [Tak19]Motoki Takigiku. A Pieri formula and a factorization formula for sums of $K$-theoretic $k$-Schur functions. Algebraic Combinatorics, 2(4):447–480, 2019.
.bib
@article{Takigiku2019, author = {Takigiku, Motoki}, title = {A {P}ieri formula and a factorization formula for sums of {$K$}-theoretic {$k$}-{S}chur functions}, year = {2019}, journal = {Algebraic Combinatorics}, volume = {2}, number = {4}, pages = {447--480}, publisher = {MathDoc/Centre Mersenne}, doi = {10.5802/alco.45}, url = {http://dx.doi.org/10.5802/alco.45}, issn = {2589-5486} }