The symmetric functions catalog

#Hall–Littlewood P

The Hall–Littlewood \(P\) polynomials interpolate between the Schur polynomials and the monomial symmetric functions. They were introduced by P. Hall (implicitly) and later by D. E. Littlewood in [Lit61]. F. Descouens and A. Lascoux define nonsymmetric Hall–Littlewood polynomial bases using Yang–Baxter elements of the Hecke algebra [DL06]. The ordinary Hall–Littlewood polynomials occur as a subfamily of one basis, and the specialization \(q=0\) recovers the two classical families of key polynomials. M. Wheeler and P. Zinn-Justin study Hall polynomials and generalized inverse Kostka polynomials using integrable lattice models and puzzle-like tilings [WZ18]. Their formulas express Hall structure constants and the mixed coefficients in \(\schurS_\mu\hallLittlewoodP_\nu\) through divided-difference operators and weighted puzzle sums; a related specialization recovers Knutson–Tao puzzles for Littlewood–Richardson coefficients. K. Motegi studies a related finite-lattice vertex-model construction based on the \(U_q(\mathfrak{sl}_2)\) \(R\)-matrix [Mot17]. The resulting wavefunctions are symmetric polynomials that specialize to Grothendieck-type polynomials, and the Yang–Baxter formalism gives determinant pairing and branching formulas. K. Chen and X. Ding give a combinatorial definition of stable spin Hall–Littlewood symmetric functions [CD21]. They prove skew Littlewood, refined Cauchy, and refined Littlewood identities, and use them to build a half-space Yang–Baxter random field. L. Petrov proves a refined Cauchy identity for fully inhomogeneous spin Hall–Littlewood symmetric rational functions [Pet21]. The identity expresses a weighted sum of products as an Izergin–Korepin determinant, links the functions to interpolation Macdonald polynomials, and has an ASEP specialization. I. Fischer and M. Gangl prove two Littlewood identities for fully inhomogeneous spin Hall–Littlewood symmetric rational functions [FG26]. One product side contains the Littlewood kernel and a simple correction factor, while the other contains the kernel and a Pfaffian. A specialization gives a new classical Hall–Littlewood identity, and the underlying path model is bijective with modified Robbins-polynomial configurations. M. Lu, S. Ruan, and W. Wang construct \(\imath\) Hall–Littlewood functions from the \(\imath\)-Hall algebra of the Jordan quiver [LRW25]. The modified \(\imath\) Hall–Littlewood functions specialize to the modified Hall–Littlewood functions at \(\theta=0\) and to deformed type \(C\) universal characters at \(\theta=1.\) J. Chen, M. Lu, and S. Ruan introduce double Hall–Littlewood symmetric polynomials via the derived Hall algebra of the Jordan quiver [CLR26]. These functions are indexed by bipartitions, reduce to ordinary Hall–Littlewood functions when one partition is empty, specialize to Schur Laurent symmetric functions at \(t=0,\) and satisfy Pieri rules. The Hall–Littlewood \(P\) polynomials also have a positive expansion in Demazure \(t\)-atoms: \[\hallLittlewoodP_\lambda(\xvec;t) = \sum_{\alpha^+=\lambda} \atom_\alpha(\xvec;t),\] where \(\alpha^+\) denotes the partition obtained by rearranging \(\alpha.\) Alexandersson’s general-basement formula refines this to a positive expansion in permuted-basement \(t\)-atoms [Ale19].

#Definition

For a partition \(\lambda,\) with \(\alpha_i\) parts equal to \(i,\) we define the Hall–Littlewood functions by \[\hallLittlewoodP_\lambda(x_1,\dotsc,x_n;t) = \frac{1}{\prod_{i\geq 0} [\alpha_i]_t! }\sum_{\sigma \in \symS_n} \sigma\left( \xvec^\lambda \frac{\prod_{i \lt j} 1-tx_j/x_i}{\prod_{i \lt j} 1-x_j/x_i} \right).\] Compare with the Weyl formula for Schur polynomials. Note that \[\hallLittlewoodP_\lambda(\xvec;1) = \monomial_\lambda(\xvec) \text{ and } \hallLittlewoodP_\lambda(\xvec;0) = \schurS_\lambda(\xvec).\] Furthermore, \(\hallLittlewoodP_\lambda(\xvec;-1)\) is equal to the Schur-\(P\) function.

The Hall–Littlewood polynomials are orthogonal with respect to the Hall–Littlewood inner product \(\langle - , - \rangle_t\) defined via \[\langle \powerSum_\lambda , \powerSum_\mu \rangle_t = \delta_{\lambda\mu} z_{\lambda} \prod_{i=1}^{\length(\lambda)} \frac{1}{1-t^{\lambda_i} },\] see [III, §4, Mac95]. This is a \(t\)-deformation of the usual Hall inner product, which is recovered at \(t=0.\)

The (skew) Hall–Littlewood \(\hallLittlewoodP\) functions can be computed via a branching rule as follows. First let the one-variable specialization be \(\hallLittlewoodP_{\lambda/\mu}(z;t) = z^{|\lambda/\mu|}\psi_{\lambda/\mu}(t),\) where \[\psi_{\lambda/\mu}(t) = \prod_{\substack{ i \geq 1 \\ m_i(\mu) = m_i(\lambda)+1 }} \left( 1-t^{m_i(\mu)} \right).\] Then the skew Hall–Littlewood \(P\) polynomial is given by \[\hallLittlewoodP_{\lambda/\mu}(x_1,\dotsc,x_n;t) \coloneqq \sum_{ \mu = \nu^{0} \prec \dotsb \prec \nu^n = \lambda} \prod_{i=1}^n \psi_{\nu^{i}/\nu^{i-1}}(t) x_i^{|\nu^{i}/\nu^{i-1}|}.\] Here, we write \(\mu \prec \lambda\) if \(\lambda/\mu\) is a skew shape.

The Hall–Littlewood \(\hallLittlewoodP\) polynomials are dual to the Hall–Littlewood \(\hallLittlewoodQ\) basis: \(\langle \hallLittlewoodQ_\lambda, \hallLittlewoodP_\mu \rangle_t = \delta_{\lambda\mu}.\)

#Cauchy identity

We have the Cauchy identity (see [Ch. III, Eq. (4.4), Mac95]), \[\prod_{i=1}^n \prod_{j=1}^n \left( \frac{1-tx_i y_j}{1-x_iy_j } \right) = \sum_{\mu} \hallLittlewoodP_\mu(x_1,\dotsc,x_n;t) \hallLittlewoodQ_\mu(y_1,\dotsc,y_n;t).\] D. Betea and M. Wheeler refine Hall–Littlewood Cauchy and Littlewood identities by inserting simple multiplicity factors depending on the parts of \(\lambda\) [BW16]. These refinements have plane-partition interpretations, while their closed forms are six-vertex model partition functions with domain-wall, off-diagonally symmetric, and reflecting boundary conditions. One refined Hall–Littlewood Cauchy identity also includes an extra parameter attached to the zero parts of \(\lambda\) and evaluates to an Izergin-type determinant. S. Ole Warnaar proves affine Jacobi–Trudi formulas for Hall–Littlewood polynomials indexed by rectangles [War25]. The identities are used to derive \(q,t\)-Rogers–Ramanujan identities and related affine Hall–Littlewood expansions.

#Hall–Littlewood–Schubert series

J. Maglione and C. Voll introduce Hall–Littlewood–Schubert series, multivariate rational generating series built from Hall–Littlewood-type tableau weights [MV24]. These series specialize to affine Schubert series for lattice enumeration, symplectic Hecke series, generalized Igusa functions, and zeta functions of certain quiver representations. They also satisfy a self-reciprocity theorem and give a Hall–Littlewood analogue of classical Littlewood identities.

R. M. Adin and T. Bauer introduce skew Hall–Littlewood–Schubert series, simultaneously refining generalized Igusa functions and Hall–Littlewood–Schubert series [AB26]. They give tableau formulations and prove a self-reciprocity functional equation by zeta and Möbius inversion on a poset of partitions.

#\(\Delta\)-Springer varieties

S. T. Griffin proves a positive Hall–Littlewood expansion formula for the graded Frobenius characteristic of the cohomology ring of a \(\Delta\)-Springer variety [Gri24]. The proof counts points over finite fields and decomposes the varieties into pieces built from Springer fibers and affine spaces. As a special case, this gives a geometric interpretation of the Hall–Littlewood expansion appearing in the Delta Conjecture at \(t=0.\)

#Hall–Littlewood \(S\)-functions

A related Jacobi–Trudi family is obtained by replacing the complete homogeneous functions by the one-row Hall–Littlewood functions. Let \(q_n(X;t)\) be defined by \[\sum_{n\geq 0} q_n(X;t)u^n = \prod_i \frac{1-tx_i u}{1-x_i u},\] and let \(\theta_t\) be the algebra homomorphism determined by \(\theta_t(h_n)=q_n(X;t).\) The Hall–Littlewood \(S\)-symmetric functions are \[S_{\lambda/\mu}(X;t) \coloneqq \theta_t(s_{\lambda/\mu}).\] Thus \(S_{\lambda/\mu}(X;0)=s_{\lambda/\mu}(X).\)

D. Grinberg and E. Vassilieva [GV24] relate this family to enriched \(P\)-partitions. If \(P_{\lambda/\mu}\) is the poset of cells in the skew shape \(\lambda/\mu,\) then they prove that \[S_{\lambda/\mu}(X;-q)=\Gamma^{(q)}(P_{\lambda/\mu}),\] where the right-hand side is the \(q\)-deformed generating function for enriched \(P\)-partitions of \(P_{\lambda/\mu}.\) They also give a \(q\)-fundamental quasisymmetric expansion indexed by standard Young tableaux of shape \(\lambda/\mu.\)

#Gelfand–Tsetlin patterns

B. Feigin and I. Makhlin give a combinatorial formula of the form \[\hallLittlewoodP_\lambda(\xvec) = \sum_{G \in \GT(\lambda) } p_G(t) \xvec^{w(G)}\] where \(p_G(t)\) is a certain polynomial defined via a statistic on the Gelfand–Tsetlin pattern \(G\) [FM16].

Another formula using GT-patterns is given in [Thm. 11, GRP15]. They generalize Tokuyama’s formula [Tok88] and recover Stanley’s formula [Sta86] for Schur \(Q\)-functions as a special case.

R. V. Peski studies the boundary of the Hall–Littlewood \(t\)-deformed Gelfand–Tsetlin graph and relates it to infinite \(p\)-adic random matrices when \(1/t\) is a prime [Pes21]. The proof uses explicit formulas for skew Hall–Littlewood polynomials and branching measures.

#Pieri rule

M. Konvalinka and A. Lauve prove a skew Pieri rule for the Hall–Littlewood \(\hallLittlewoodP\) functions [KL12]. Below is the simpler nonskew version.

#Kostka–Foulkes polynomials

The polynomials \(K_{\lambda\mu}(t)\) in the expansion \[\schurS_\lambda(\xvec) = \sum_\mu K_{\lambda\mu}(t) \hallLittlewoodP_\mu(\xvec)\] are called Kostka–Foulkes polynomials and are certain \(t\)-analogues of the Kostka coefficients. A. Lascoux and M.-P. Schützenberger prove that \[K_{\lambda\mu}(t) = \sum_{T \in \mathrm{SSYT}(\lambda,\mu)} t^{\charge(T)},\] where the sum is over semistandard Young tableaux and charge is a certain statistic on SSYTs [LS78]. J. Carbonara gives a combinatorial interpretation for the inverse \(t\)-Kostka matrix, the transition matrix from Hall–Littlewood polynomials back to Schur functions [Car98]. The model uses tournament matrices and gives a \(t\)-analogue of the inverse Kostka interpretation by special rim hook tabloids. Kostka–Foulkes polynomials can also be defined using rigged configurations, introduced by A. Kirillov and N. Reshetikhin [KR88]. See the page on Kostka–Foulkes polynomials for more information.

#Products

There are various products that lead to \(t\)-deformations of known structure constants.

#Hall polynomials

The Hall polynomials, \(c^{\nu}_{\lambda\mu}(t) \in \setZ[t]\) deform the Littlewood–Richardson coefficients. They are defined via the expansion \[\hallLittlewoodP_\lambda(\xvec) \hallLittlewoodP_\mu(\xvec) = \sum_\nu c^{\nu}_{\lambda\mu}(t) \hallLittlewoodP_\nu(\xvec),\] and at \(t=0,\) the Littlewood–Richardson coefficients for Schur polynomials are recovered. Note that the polynomials \(c^{\nu}_{\lambda\mu}(t)\) may have negative coefficients.

A combinatorial model for \(c^{\nu}_{\lambda\mu}(t)\) can be recovered from the more general theory in [Yip12]. An earlier proof appears in [Sch06].

P. Zinn-Justin describes a honeycomb model for the Hall polynomials [Zin19]. This extends the honeycomb model by Knutson–Tao for the Littlewood–Richardson coefficients. A. Gunna, M. Wheeler, and P. Zinn-Justin give a combinatorial formula for structure constants of spin Hall–Littlewood functions [GWZ25]. Their formula comes from a lattice-model partition function and the Yang–Baxter equation, and is expressed using generalized honeycombs.

The Hall polynomials have significance in the study of abelian \(p\)-groups. Let \(p\) be a prime number. An abelian \(p\)-group of type \(\lambda\) is isomorphic to \[G = \setZ/p^{\lambda_1} \oplus \setZ/p^{\lambda_2} \oplus \dotsb \oplus \setZ/p^{\lambda_\ell}.\] A subgroup \(H \subseteq G\) has cotype \(\nu\) if \(G/H\) has type \(\nu.\)

For fixed \(\lambda,\) the Hall polynomial \(c^{\lambda}_{\mu\nu}(p)\) has nonnegative coefficients for all \(\mu\) and \(\nu\) if and only if no two parts of \(\lambda\) differ by more than one.

L. M. Butler and A. W. Hales determine exactly the parameters for which all coefficients are positive [BH93]. This result later inspired a proof of the following theorem.

#Other deformations

M. Wheeler and P. Zinn-Justin give a signed combinatorial formula for \(\overline{K}^{\nu}_{\lambda\mu}(t)\) in the product [Thm. 4, WZ18] \[\hallLittlewoodP_\lambda(\xvec;t) \schurS_\mu(\xvec) = \sum_{\nu} \overline{K}^{\nu}_{\lambda\mu}(t) \schurS_\nu(\xvec).\]

#Conjectures

#MathOverflow conjecture

Let \(\lambda = (\lambda_1,\dotsc,\lambda_\ell)\) be a partition satisfying \(\lambda_i \geq \lambda_{i+1}+2\) for all \(1 \leq i \leq \ell-1.\) In the MathOverflow post [Mak], it is conjectured that \(\hallLittlewoodP_\lambda(\xvec,-t)\) expands positively in the Schur polynomial basis under this condition on \(\lambda.\) This conjecture should be attributed to I. Makhlin (monomial positivity) and R. Stanley (Schur positivity).

#Bhattacharya conjecture

A. Bhattacharya (personal communication, FPSAC 2022 Bangalore) suggests the following conjecture:

Note that the first conjecture implies that expanding \(\hallLittlewoodP_\lambda(\xvec,t)\) into \(\hallLittlewoodP_\mu(\xvec,t+m)\) for any \(m \geq 1\) also gives nonnegative coefficients.

This conjecture does not generalize to quasisymmetric Hall–Littlewood polynomials; a counterexample exists for \(n=5.\) For the definition of quasisymmetric Schur functions, see [HLMW11].

#Hall–Littlewood Q

The Hall–Littlewood \(Q\) polynomials can be defined as \[\hallLittlewoodQ_\lambda(x_1,\dotsc,x_n;t) = (1-t)^{\length(\lambda)}[n-\length(\lambda)]_t! \sum_{\sigma \in \symS_n} \sigma\left( \xvec^\lambda \frac{\prod_{i \lt j} 1-tx_j/x_i}{\prod_{i \lt j} 1-x_j/x_i} \right).\] Thus, \(\hallLittlewoodQ_\lambda(\xvec;t)\) and \(\hallLittlewoodP_\lambda(\xvec;t)\) differ only by a factor in \(\setZ[t].\) See [III. §2, Mac95] for this normalization. More explicitly, \(\hallLittlewoodQ_\lambda=b_\lambda(t)\hallLittlewoodP_\lambda,\) where \[b_\lambda(t)=\prod_{i\geq 1}\prod_{j=1}^{m_i(\lambda)}(1-t^j).\] In particular, \(\hallLittlewoodQ_{(1)}=(1-t)\schurS_{(1)}.\) Thus this \(Q\)-normalization is not Schur-positive over \(\setN[t],\) and the specialization \(t=1\) is not the complete homogeneous basis. The Schur-positive Hall–Littlewood family governed by Kostka–Foulkes polynomials is instead the transformed or modified Hall–Littlewood family.

See also [Jin91] for a similar way of creating Hall–Littlewood functions via a modification of the Bernstein operator.

J. Graf constructs a \(t\)-analogue of the Bernstein operator which gives an explicit construction of Jing’s Hall–Littlewood vertex operator [Gra25]. The construction emphasizes the role of the involution \(\omega\) and is applied to stability of structure coefficients, including Hall polynomials.

#Transformed Hall–Littlewood

The transformed Hall–Littlewood polynomials are defined via the Kostka–Foulkes polynomials as \[\hallLittlewoodT_\mu(\xvec;q) \coloneqq \sum_{\lambda} K_{\lambda\mu}(q) \schurS_\lambda(\xvec).\] These are sometimes also denoted \(Q'_\mu(\xvec;q)\); see [DLT94, TZ03]. One has \[\hallLittlewoodT_\mu(\xvec;q) = \prod_{i \lt j} \frac{1-R_{ij}}{1-q R_{ij}} \completeH_{\mu}(\xvec) = \prod_{i \lt j} (1-q R_{ij})^{-1} \schurS_{\mu}(\xvec)\] where we use the same notation as in the raising operator formula for Schur polynomials. In particular, from this definition it is clear that \(\hallLittlewoodT_\mu(\xvec;0) = \schurS_\mu(\xvec).\)

The relationship between the classical and transformed Hall–Littlewood functions is given by the plethystic relationship \[\hallLittlewoodQ_\mu(\xvec;q) = \hallLittlewoodT_\mu[(1-q)\xvec;q].\]

G. Tudose and M. Zabrocki introduce a \(q\)-deformation of Schur’s \(Q\)-functions in a manner analogous to the construction above [TZ03].

The transformed Hall–Littlewood polynomials are special cases of the Catalan symmetric functions. These also define compositional Hall–Littlewood polynomials \(\hallLittlewoodT_{\gamma}(\xvec;q),\) where \(\gamma\) is a composition.

We also have the following relation with nonsymmetric Macdonald polynomials: \[\omega\hallLittlewoodT_{\lambda'}(x_1,\dotsc,x_n;q) = \macdonaldE_{\rev(\lambda)}(x_1,\dotsc,x_n;q,0).\] A consequence of this relation is the following identity involving LLT polynomials: \[q^{\sum_{i\geq 2} \binom{\lambda_i}{2} } \omega \hallLittlewoodH_\lambda(\xvec;t) = \LLT_{\nuvec}(\xvec;q)\] where \(\nuvec = (1^{\lambda_1}, 1^{\lambda_2},\dotsc,1^{\lambda_\ell}).\)

#Modified Hall–Littlewood polynomials

The modified Hall–Littlewood polynomials are related to the transformed Hall–Littlewood polynomials via \[\hallLittlewoodH_\lambda(\xvec;t) = t^{\partitionN(\lambda)} \hallLittlewoodT_\lambda(\xvec;t^{-1}).\] Equivalently, \[\hallLittlewoodH_\mu(\xvec;t) = \sum_{\lambda}\tilde{K}_{\lambda\mu}(t)\schurS_\lambda(\xvec).\] Since \(\tilde{K}_{\mu\mu}(t)=1\) and \(\tilde{K}_{\lambda\mu}(t)=0\) unless \(\lambda\) dominates \(\mu,\) this gives a unitriangular Schur expansion. Thus the modified Hall–Littlewood basis is uniquely determined by these coefficients. The modified Hall–Littlewood polynomials have a direct combinatorial Schur expansion. They are Schur-positive and are in fact the Frobenius image of a certain graded vector space.

The modified Hall–Littlewood polynomials are specializations of the modified Macdonald polynomials.

#Modified Kostka–Foulkes polynomials

The modified Kostka–Foulkes polynomials are defined as \[\tilde{K}_{\lambda\mu}(t) = t^{\partitionN(\mu)}K_{\lambda\mu}(1/t) = \sum_{T \in \mathrm{SSYT}(\lambda,\mu)} t^{\cocharge(T)},\] where the sum is over semistandard Young tableaux and cocharge is defined via the charge statistic.

These are precisely the Schur expansion coefficients for \(\hallLittlewoodH_\mu(\xvec;t)\) in the formula above.

#In other root systems

The symmetric functions above are the type \(A\) Hall–Littlewood polynomials. There are analogous Hall–Littlewood polynomials for other reduced root systems, where partitions are replaced by dominant weights and the Weyl-group symmetrization uses the positive roots of the root system; see [Ch. V, §5, Mac95]. From the Macdonald-polynomial point of view, these are the \(q=0\) specialization of the root-system Macdonald polynomials.

There are also Kac–Moody analogues. In this setting the same kind of Weyl-symmetrized expression has to be interpreted with care because the root system is infinite; for a short introduction, see S. Viswanath’s article [Vis06].

Bibliography

1. [AB26]Ron M. Adin and Tomer Bauer. Reciprocity of Skew Hall-Littlewood-Schubert Series. arXiv:2603.29728v1, 2026.
   .bib

   Copy Download

   @article{AdinBauer2026x,
     author = {Ron M. Adin and Tomer Bauer},
     title = {Reciprocity of {S}kew {H}all-{L}ittlewood-{S}chubert {S}eries},
     year = {2026},
     eprint = {2603.29728v1},
     url = {https://arxiv.org/abs/2603.29728v1},
     journal = {arXiv e-prints}
   }
   

2. [Ale19]Per Alexandersson. Non-symmetric Macdonald polynomials and Demazure–Lusztig operators. Séminaire Lotharingien de Combinatoire, 76, 2019.
   .bib

   Copy Download

   @article{Alexandersson2015gbMacdonald,
     author = {Per Alexandersson},
     title = {Non-symmetric {M}acdonald polynomials and {D}emazure--{L}usztig operators},
     journal = {Séminaire Lotharingien de Combinatoire},
     volume = {76},
     year = {2019},
     url = {https://www.mat.univie.ac.at/~slc/wpapers/s76alexand.html}
   }
   

3. [Ale20]Per Alexandersson. LLT polynomials, elementary symmetric functions and melting lollipops. Journal of Algebraic Combinatorics, 53(2):299–325, April 2020.
   .bib

   Copy Download

   @article{Alexandersson2019llt,
     doi = {10.1007/s10801-019-00929-z},
     url2 = {https://doi.org/10.1007/s10801-019-00929-z},
     year = {2020},
     month = apr,
     publisher = {Springer Science and Business Media {LLC}},
     volume = {53},
     number = {2},
     pages = {299--325},
     author = {Per Alexandersson},
     title = {{LLT} polynomials,  elementary symmetric functions and melting lollipops},
     journal = {Journal of Algebraic Combinatorics}
   }
   

4. [AS20]Per Alexandersson and Robin Sulzgruber. A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions. arXiv:2004.09198, 2020.
   .bib

   Copy Download

   @article{AlexanderssonSulzgruber2020x,
   Author = {Per Alexandersson and Robin Sulzgruber},
   Title = {A combinatorial expansion of vertical-strip {LLT} polynomials in the basis of elementary symmetric functions},
   Year = {2020},
   Eprint = {2004.09198},
   url = {https://arxiv.org/abs/2004.09198},
   journal = {arXiv e-prints}
   }
   

5. [BW16]D. Betea and M. Wheeler. Refined Cauchy and Littlewood identities, plane partitions and symmetry classes of alternating sign matrices. Journal of Combinatorial Theory, Series A, 137:126–165, January 2016.
   .bib

   Copy Download

   @article{BeteaWheeler2016,
     doi = {10.1016/j.jcta.2015.08.007},
     url2 = {https://doi.org/10.1016/j.jcta.2015.08.007},
     year = {2016},
     month = jan,
     publisher = {Elsevier {BV}},
     volume = {137},
     pages = {126--165},
     author = {D. Betea and M. Wheeler},
     title = {Refined {C}auchy and {L}ittlewood identities,  plane partitions and symmetry classes of alternating sign matrices},
     journal = {Journal of Combinatorial Theory,  Series A}
   }
   

6. [BH93]Lynne M. Butler and Alfred W. Hales. Nonnegative Hall polynomials. Journal of Algebraic Combinatorics, 2(2):125–135, 1993.
   .bib

   Copy Download

   @article{ButlerHales1993,
     doi = {10.1023/a:1022407523839},
     url2 = {https://doi.org/10.1023/a:1022407523839},
     year = {1993},
     publisher = {Springer Nature},
     volume = {2},
     number = {2},
     pages = {125--135},
     title = {Nonnegative {H}all polynomials},
     author = {Lynne M. Butler and Alfred W. Hales},
     journal = {Journal of Algebraic Combinatorics}
   }
   

7. [Car98]Joaquin O. Carbonara. A combinatorial interpretation of the inverse t-Kostka matrix. Discrete Mathematics, 193(1-3):117–145, November 1998.
   .bib

   Copy Download

   @article{Carbonara1998,
     doi = {10.1016/s0012-365x(98)00138-1},
     url2 = {https://doi.org/10.1016/s0012-365x(98)00138-1},
     year = {1998},
     month = nov,
     publisher = {Elsevier {BV}},
     volume = {193},
     number = {1-3},
     pages = {117--145},
     author = {Joaquin O. Carbonara},
     title = {A combinatorial interpretation of the inverse t-{K}ostka matrix},
     journal = {Discrete Mathematics}
   }
   

8. [CD21]Kailun Chen and Xiangmao Ding. Stable spin Hall–Littlewood symmetric functions, combinatorial identities, and half-space Yang–Baxter random field. arXiv:2106.12557, 2021.
   .bib

   Copy Download

   @article{ChenDing2021SpinHL,
     author = {Kailun Chen and Xiangmao Ding},
     title = {Stable spin {H}all--{L}ittlewood symmetric functions,
       combinatorial identities, and half-space {Y}ang--{B}axter random field},
     year = {2021},
     eprint = {2106.12557},
     url = {https://arxiv.org/abs/2106.12557},
     journal = {arXiv e-prints}
   }
   

9. [CLR26]Jiayi Chen, Ming Lu and Shiquan Ruan. Double Hall-Littlewood symmetric polynomials. arXiv:2601.13497, 2026.
   .bib

   Copy Download

   @article{ChenLuRuan2026x,
     author = {Jiayi Chen and Ming Lu and Shiquan Ruan},
     title = {Double {H}all-{L}ittlewood symmetric polynomials},
     year = {2026},
     eprint = {2601.13497},
     url = {https://arxiv.org/abs/2601.13497},
     journal = {arXiv e-prints}
   }
   

10. [DLT94]Jacques Désarménien, Bernard Leclerc and Jean-Yves Thibon. Hall-Littlewood functions and Kostka–Foulkes polynomials in representation theory. Séminaire Lotharingien de Combinatoire [electronic only], 32:38, 1994.
    .bib

    Copy Download

    @article{DesarmenienLeclercThibon1994,
    author = {Jacques D{\'{e}}sarm{\'{e}}nien and Bernard Leclerc and Jean-Yves Thibon},
    journal = {S{\'{e}}minaire Lotharingien de Combinatoire [electronic only]},
    language = {eng},
    pages = {38},
    publisher = {Universit{\"{a}}t Wien, Fakult{\"{a}}t f{\"{u}}r Mathematik},
    title = {Hall-{L}ittlewood functions and {K}ostka--{F}oulkes polynomials in representation theory},
    url = {http://eudml.org/doc/119019},
    volume = {32},
    year = {1994},
    }
    

11. [DL06]Francois Descouens and Alain Lascoux. Non-Symmetric Hall–Littlewood Polynomials. arXiv:math/0609512, 2006.
    .bib

    Copy Download

    @article{DescouensLascoux2006x,
      author = {Francois Descouens and Alain Lascoux},
      title = {Non-{S}ymmetric {H}all--{L}ittlewood {P}olynomials},
      year = {2006},
      eprint = {math/0609512},
      url = {https://arxiv.org/abs/math/0609512},
      journal = {arXiv e-prints}
    }
    

12. [FM16]Boris Feigin and Igor Makhlin. A combinatorial formula for affine Hall–Littlewood functions via a weighted Brion theorem. Selecta Mathematica, 22(3):1703–1747, March 2016.
    .bib

    Copy Download

    @article{FeiginMakhlin2016,
      doi = {10.1007/s00029-016-0223-4},
      url2 = {https://doi.org/10.1007/s00029-016-0223-4},
      year  = {2016},
      month = mar,
      publisher = {Springer Nature},
      volume = {22},
      number = {3},
      pages = {1703--1747},
      author = {Boris Feigin and Igor Makhlin},
      title = {A combinatorial formula for affine {H}all--{L}ittlewood functions via a weighted {B}rion theorem},
      journal = {Selecta Mathematica}
    }
    

13. [FG26]Ilse Fischer and Moritz Gangl. Two Littlewood identities for fully inhomogeneous spin Hall-Littlewood symmetric rational functions. arXiv:2603.29836v1, 2026.
    .bib

    Copy Download

    @article{FischerGangl2026x,
      author = {Ilse Fischer and Moritz Gangl},
      title = {Two {L}ittlewood identities for fully inhomogeneous spin
        {H}all-{L}ittlewood symmetric rational functions},
      year = {2026},
      eprint = {2603.29836v1},
      url = {https://arxiv.org/abs/2603.29836v1},
      journal = {arXiv e-prints}
    }
    

14. [Gra25]John Graf. Constructing Hall-Littlewood Functions via a Deformation of the Bernstein Operator. arXiv:2511.01114, 2025.
    .bib

    Copy Download

    @article{Graf2025x,
      author = {John Graf},
      title = {Constructing {H}all-{L}ittlewood {F}unctions via a {D}eformation of
        the {B}ernstein {O}perator},
      year = {2025},
      eprint = {2511.01114},
      url = {https://arxiv.org/abs/2511.01114},
      journal = {arXiv e-prints}
    }
    

15. [Gri24]Sean T. Griffin. $\Delta$-Springer varieties and Hall–Littlewood polynomials. Forum of Mathematics, Sigma, 12, 2024.
    .bib

    Copy Download

    @article{Griffin2024,
      author = {Sean T. Griffin},
      title = {{$\Delta$-Springer varieties and Hall--Littlewood polynomials}},
      year = {2024},
      journal = {Forum of Mathematics, Sigma},
      volume = {12},
      doi = {10.1017/fms.2024.1},
      url = {https://doi.org/10.1017/fms.2024.1},
      eprint = {2209.03503}
    }
    

16. [GV24]Darij Grinberg and Ekaterina A. Vassilieva. Quasisymmetric expansion of Hall-Littlewood symmetric functions. arXiv:2406.01166, 2024.
    .bib

    Copy Download

    @article{GrinbergVassilieva2024x,
      author = {Darij Grinberg and Ekaterina A. Vassilieva},
      title = {Quasisymmetric expansion of {H}all-{L}ittlewood symmetric functions},
      year = {2024},
      eprint = {2406.01166},
      url = {https://arxiv.org/abs/2406.01166},
      journal = {arXiv e-prints},
      journalref = {Séminaire Lotharingien de Combinatoire 91B (2024),
        Article \#86}
    }
    

17. [GWZ25]Ajeeth Gunna, Michael Wheeler and Paul Zinn-Justin. Structure constants for spin Hall–Littlewood functions. arXiv:2504.19205, 2025.
    .bib

    Copy Download

    @article{GunnaWheelerZinnJustin2025SpinHL,
      author = {Ajeeth Gunna and Michael Wheeler and Paul Zinn-Justin},
      title = {Structure constants for spin {H}all--{L}ittlewood functions},
      year = {2025},
      eprint = {2504.19205},
      url = {https://arxiv.org/abs/2504.19205},
      journal = {arXiv e-prints}
    }
    

18. [GRP15]Vineet Gupta, Uma Roy and Roger Van Peski. A generalization of Tokuyama’s formula to the Hall–Littlewood polynomials. The Electronic Journal of Combinatorics, 22(2), April 2015.
    .bib

    Copy Download

    @article{GuptaRoyPeski2015,
    	doi = {10.37236/4732},
    	url2 = {https://doi.org/10.37236%2F4732},
    	year = 2015,
    	month = {apr},
    	publisher = {The Electronic Journal of Combinatorics},
    	volume = {22},
    	number = {2},
    	author = {Vineet Gupta and Uma Roy and Roger Van Peski},
    	title = {A Generalization of {T}okuyama's Formula to the {H}all--{L}ittlewood Polynomials},
    	journal = {The Electronic Journal of Combinatorics}
    }
    

19. [HLMW11]James Haglund, Kurt W. Luoto, Sarah K. Mason and Stephanie Willigenburg. Quasisymmetric Schur functions. Journal of Combinatorial Theory, Series A, 118(2):463–490, 2011.
    .bib

    Copy Download

    @article{HaglundLuotoMasonWilligenburg2011,
    title = {Quasisymmetric {S}chur functions},
    journal = {Journal of Combinatorial Theory, Series A},
    volume = {118},
    number = {2},
    pages = {463--490},
    year = {2011},
    issn = {0097-3165},
    doi = {10.1016/j.jcta.2009.11.002},
    url2 = {http://www.sciencedirect.com/science/article/pii/S0097316509001745},
    author = {James Haglund and Kurt W. Luoto and Sarah K. Mason and Stephanie {van Willigenburg}}
    }
    

20. [Jin91]Naihuan Jing. Vertex operators and Hall–Littlewood symmetric functions. Advances in Mathematics, 87(2):226–248, June 1991.
    .bib

    Copy Download

    @article{Jing1991,
      title = {Vertex operators and {H}all--{L}ittlewood symmetric functions},
      volume = {87},
      ISSN = {0001-8708},
      url = {http://dx.doi.org/10.1016/0001-8708(91)90072-F},
      DOI = {10.1016/0001-8708(91)90072-f},
      number = {2},
      journal = {Advances in Mathematics},
      publisher = {Elsevier BV},
      author = {Jing,  Naihuan},
      year = {1991},
      month = jun,
      pages = {226–248}
    }
    

21. [KR88]A. N. Kirillov and N. Yu. Reshetikhin. The Bethe ansatz and the combinatorics of Young tableaux. Journal of Mathematical Sciences, 41:925–955, 1988.
    .bib

    Copy Download

    @article{KirillovReshetikhin1988thebethe,
      author = {A. N. Kirillov and N. Yu. Reshetikhin},
      title = {The {B}ethe Ansatz and the combinatorics of {Y}oung tableaux},
      journal = {Journal of Mathematical Sciences},
      year = {1988},
      volume = {41},
      pages = {925--955},
      issn = {1072-3374},
      issue = {2},
      keyword = {Mathematics and Statistics},
      publisher = {Springer New York},
      doi = {10.1007/BF01247088}
    }
    

22. [KL12]Matjaž Konvalinka and Aaron Lauve. Skew Pieri rules for Hall–Littlewood functions. Journal of Algebraic Combinatorics, 38(3):499–518, August 2012.
    .bib

    Copy Download

    @article{KonvalinkaLauve2012,
      doi = {10.1007/s10801-012-0390-0},
      url2 = {https://doi.org/10.1007/s10801-012-0390-0},
      year = {2012},
      month = aug,
      publisher = {Springer Science and Business Media {LLC}},
      volume = {38},
      number = {3},
      pages = {499--518},
      author = {Matja{\v{z}} Konvalinka and Aaron Lauve},
      title = {Skew {P}ieri rules for {H}all--{L}ittlewood functions},
      journal = {Journal of Algebraic Combinatorics}
    }
    

23. [LS78]Alain Lascoux and Marcel-Paul Schützenberger. Sur une conjecture de H. O. Foulkes. C. R. Acad. Sci. Paris Sér. A-B, 286(7):A323–A324, 1978.
    .bib

    Copy Download

    @article{LascouxSchutzenberger78,
        AUTHOR = {Lascoux, Alain and Sch{\"{u}}tzenberger, Marcel-Paul},
         TITLE = {Sur une conjecture de {H}. {O}. {F}oulkes},
       JOURNAL = {C. R. Acad. Sci. Paris S{\'{e}}r. A-B},
        VOLUME = {286},
          YEAR = {1978},
        NUMBER = {7},
         PAGES = {A323--A324}
    }
    

24. [Lit61]D. E. Littlewood. On certain symmetric functions. Proceedings of the London Mathematical Society, s3-11(1):485–498, 1961.
    .bib

    Copy Download

    @article{Littlewood1961,
      doi = {10.1112/plms/s3-11.1.485},
      url2 = {https://doi.org/10.1112/plms/s3-11.1.485},
      year = {1961},
      publisher = {Wiley},
      volume = {s3-11},
      number = {1},
      pages = {485--498},
      author = {D. E. Littlewood},
      title = {On Certain Symmetric Functions},
      journal = {Proceedings of the London Mathematical Society}
    }
    

25. [LRW25]Ming Lu, Shiquan Ruan and Weiqiang Wang. $\imath$Hall algebra of Jordan quiver and $\imath$Hall–Littlewood functions. Selecta Mathematica, 31(5), 2025.
    .bib

    Copy Download

    @article{LuRuanWang2025,
      author = {Lu, Ming and Ruan, Shiquan and Wang, Weiqiang},
      title = {{$\imath$}{H}all algebra of {J}ordan quiver and
        {$\imath$}{H}all--{L}ittlewood functions},
      year = {2025},
      journal = {Selecta Mathematica},
      volume = {31},
      number = {5},
      publisher = {Springer Science and Business Media LLC},
      doi = {10.1007/s00029-025-01101-1},
      url = {http://dx.doi.org/10.1007/s00029-025-01101-1},
      issn = {1420-9020},
      eprint = {2104.12336}
    }
    

26. [Mak]Igor Makhlin. On a positivity property of Hall–Littlewood polynomials. MathOverflow, . URL:https://mathoverflow.net/q/190646 (version: 2014-12-17)
    .bib

    Copy Download

    @MISC{MO190646,
        TITLE = {On a positivity property of {H}all--{L}ittlewood polynomials},
        AUTHOR = {Igor Makhlin},
        HOWPUBLISHED = {MathOverflow},
        NOTE = {URL:https://mathoverflow.net/q/190646 (version: 2014-12-17)},
        EPRINT = {https://mathoverflow.net/q/190646},
        URL = {https://mathoverflow.net/q/190646}
    }
    

27. [Mac95]Ian G. Macdonald. Symmetric functions and Hall polynomials. Oxford mathematical monographs. The Clarendon Press, Oxford University Press, 1995. With contributions by A. Zelevinsky, Oxford Science Publications
    .bib

    Copy Download

    @book{Macdonald1995,
    	address = {New York},
    	author = {Ian G. Macdonald},
    	edition = {Second},
    	isbn = {0-19-853489-2},
    	mrclass = {05E05 (05-02 20C30 20C33 20K01 33C80 33D80)},
    	note = {With contributions by A. Zelevinsky, Oxford Science Publications},
    	pages = {x+475},
    	publisher = {The Clarendon Press, Oxford University Press},
    	series = {Oxford Mathematical Monographs},
    	title = {Symmetric functions and {H}all polynomials},
    	year = {1995}
    }
    

28. [MV24]Joshua Maglione and Christopher Voll. Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration. arXiv:2410.08075, 2024.
    .bib

    Copy Download

    @article{MaglioneVoll2024x,
      author = {Joshua Maglione and Christopher Voll},
      title = {Hall-{L}ittlewood polynomials, affine {S}chubert series, and lattice enumeration},
      year = {2024},
      eprint = {2410.08075},
      url = {https://arxiv.org/abs/2410.08075},
      journal = {arXiv e-prints}
    }
    

29. [Mal96]F. Miller Maley. The Hall polynomial revisited. Journal of Algebra, 184(2):363–371, September 1996.
    .bib

    Copy Download

    @article{Maley1996,
      doi = {10.1006/jabr.1996.0264},
      url2 = {https://doi.org/10.1006/jabr.1996.0264},
      year = {1996},
      month = sep,
      publisher = {Elsevier {BV}},
      volume = {184},
      number = {2},
      pages = {363--371},
      author = {F. Miller Maley},
      title = {The {H}all Polynomial Revisited},
      journal = {Journal of Algebra}
    }
    

30. [Mot17]Kohei Motegi. Combinatorial properties of symmetric polynomials from integrable vertex models in finite lattice. Journal of Mathematical Physics, 58(9), 2017.
    .bib

    Copy Download

    @article{Motegi2017,
      author = {Motegi, Kohei},
      title = {Combinatorial properties of symmetric polynomials from integrable
        vertex models in finite lattice},
      year = {2017},
      journal = {Journal of Mathematical Physics},
      volume = {58},
      number = {9},
      publisher = {AIP Publishing},
      doi = {10.1063/1.5001687},
      url = {http://dx.doi.org/10.1063/1.5001687},
      issn = {1089-7658}
    }
    

31. [Pes21]Roger Van Peski. Hall–Littlewood polynomials, boundaries, and $p$-adic random matrices. arXiv:2112.02147, 2021.
    .bib

    Copy Download

    @article{Peski2021x,
      author = {Roger Van Peski},
      title = {Hall--{L}ittlewood polynomials, boundaries, and $p$-adic random
        matrices},
      year = {2021},
      eprint = {2112.02147},
      url = {https://arxiv.org/abs/2112.02147},
      journal = {arXiv e-prints}
    }
    

32. [Pet21]Leonid Petrov. Refined Cauchy identity for spin Hall–Littlewood symmetric rational functions. Journal of Combinatorial Theory, Series A, 184:105519, 2021.
    .bib

    Copy Download

    @article{Petrov2021SpinHLCauchy,
      author = {Petrov, Leonid},
      title = {Refined {C}auchy identity for spin {H}all--{L}ittlewood symmetric
        rational functions},
      year = {2021},
      journal = {Journal of Combinatorial Theory, Series A},
      volume = {184},
      pages = {105519},
      publisher = {Elsevier BV},
      doi = {10.1016/j.jcta.2021.105519},
      url = {http://dx.doi.org/10.1016/j.jcta.2021.105519},
      issn = {0097-3165},
      eprint = {2007.10886}
    }
    

33. [Sch06]C. Schwer. Galleries, Hall–Littlewood polynomials, and structure constants of the spherical Hecke algebra. International Mathematics Research Notices, January 2006.
    .bib

    Copy Download

    @article{Schwer2006,
      doi = {10.1155/imrn/2006/75395},
      url2 = {https://doi.org/10.1155/imrn/2006/75395},
      year = {2006},
      month = jan,
      publisher = {Oxford University Press ({OUP})},
      author = {C. Schwer},
      title = {Galleries,  {H}all--{L}ittlewood polynomials,  and structure constants of the spherical {H}ecke algebra},
      journal = {International Mathematics Research Notices}
    }
    

34. [Sta86]Richard P. Stanley. A Baker’s dozen of conjectures concerning plane partitions. Combinatoire énumérative:285–293, 1986.
    .bib

    Copy Download

    @inproceedings{Stanley1986,
        doi = {10.1007/BFb0072521},
        author={Richard P. Stanley},
        editor={Labelle, Gilbert and Leroux, Pierre},
        title={A {B}aker's dozen of conjectures concerning plane partitions},
        booktitle={Combinatoire {\'{e}}num{\'{e}}rative},
        year={1986},
        publisher={Springer Berlin Heidelberg},
        address={Berlin, Heidelberg},
        pages={285--293},
        isbn={978-3-540-47402-9}
    }
    

35. [Tok88]Takeshi Tokuyama. A generating function of strict Gelfand patterns and some formulas on characters of general linear groups. Journal of the Mathematical Society of Japan, 40(4):671–685, October 1988.
    .bib

    Copy Download

    @article{Tokuyama1988,
    author = {Takeshi Tokuyama},
    doi = {10.2969/jmsj/04040671},
    journal = {Journal of the Mathematical Society of Japan},
    month = oct,
    number = {4},
    pages = {671--685},
    publisher = {Mathematical Society of Japan},
    title = {A generating function of strict {G}elfand patterns and some formulas on characters of general linear groups},
    volume = {40},
    year = {1988}
    }
    

36. [TZ03]Geanina Tudose and Mike Zabrocki. A $q$-analog of Schur’s Q-functions. Algebraic combinatorics and quantum groups, 2003.
    .bib

    Copy Download

    @incollection{TudoseZabrocki2003,
      author = {Geanina Tudose and Mike Zabrocki},
      title = {A $q$-Analog of {S}chur's {Q}-Functions},
      booktitle = {Algebraic Combinatorics and Quantum Groups},
      publisher = {World Scientific},
      year = {2003},
      editor = {Naihuan Jing},
      doi = {10.1142/5331}
    }
    

37. [Vis06]Sankaran Viswanath. Kostka-Foulkes polynomials for symmetrizable Kac-Moody algebras. arXiv:math/0610246, 2006.
    .bib

    Copy Download

    @article{Viswanath2006x,
      author = {Sankaran Viswanath},
      title = {Kostka-{F}oulkes polynomials for symmetrizable {K}ac-{M}oody algebras},
      year = {2006},
      eprint = {math/0610246},
      url = {https://arxiv.org/abs/math/0610246},
      journal = {arXiv e-prints},
      journalref = {Seminaire Lotharingien de Combinatoire, 58 (2008), Article B58f}
    }
    

38. [War25]S. Ole Warnaar. Affine Jacobi-Trudi formulas and $q,t$-Rogers-Ramanujan identities. arXiv:2511.17034, 2025.
    .bib

    Copy Download

    @article{Warnaar2025x,
      author = {S. Ole Warnaar},
      title = {Affine {J}acobi-{T}rudi formulas and $q,t$-{R}ogers-{R}amanujan
        identities},
      year = {2025},
      eprint = {2511.17034},
      url = {https://arxiv.org/abs/2511.17034},
      journal = {arXiv e-prints}
    }
    

39. [WZ18]M. Wheeler and P. Zinn-Justin. Hall polynomials, inverse Kostka polynomials and puzzles. Journal of Combinatorial Theory, Series A, 159:107–163, October 2018.
    .bib

    Copy Download

    @article{WheelerZinnJustin2018,
      doi = {10.1016/j.jcta.2018.05.005},
      url2 = {https://doi.org/10.1016/j.jcta.2018.05.005},
      year = {2018},
      month = oct,
      publisher = {Elsevier {BV}},
      volume = {159},
      pages = {107--163},
      author = {M. Wheeler and P. Zinn-Justin},
      title = {{H}all polynomials,  inverse {K}ostka polynomials and puzzles},
      journal = {Journal of Combinatorial Theory,  Series A}
    }
    

40. [Yip12]Martha Yip. A Littlewood–Richardson rule for Macdonald polynomials. Mathematische Zeitschrift, 272(3-4):1259–1290, March 2012.
    .bib

    Copy Download

    @article{Yip2012,
      doi = {10.1007/s00209-012-0986-z},
      url2 = {https://doi.org/10.1007/s00209-012-0986-z},
      year = {2012},
      month = mar,
      publisher = {Springer Science and Business Media {LLC}},
      volume = {272},
      number = {3-4},
      pages = {1259--1290},
      author = {Martha Yip},
      title = {A {L}ittlewood--{R}ichardson rule for {M}acdonald polynomials},
      journal = {Mathematische Zeitschrift}
    }
    

41. [Zin19]Paul Zinn-Justin. Honeycombs for Hall polynomials. arXiv:1909.10720, 2019.
    .bib

    Copy Download

    @article{ZinnJustin2019,
    Author = {Paul Zinn-Justin},
    Title = {Honeycombs for {H}all polynomials},
    Year = {2019},
    Eprint = {1909.10720},
      url = {https://arxiv.org/abs/1909.10720},
    journal = {arXiv e-prints}
    }