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#Shifted Schur polynomials

Shifted Schur functions were studied in [OO96] and are closely related to the factorial Schur polynomials. The shifted Schur functions are specializations of the shifted Jack functions.

J. Gao, S. Gao, and Y. Gao introduce balanced shifted tableaux as type \(B/C\) analogues of Edelman–Greene balanced tableaux [GGG24]. They give an explicit bijection with standard Young tableaux of the same shifted shape. A. Brini and A. Teolis identify certain central Schur elements in \(\mathbf{U}(\mathfrak{gl}_n)\) with the preimages of shifted Schur polynomials under the Harish–Chandra isomorphism [BT22]. Equivalently, these elements recover Okounkov’s quantum immanants.

Define the falling factorial as \[\fallingFactorial{x}{k} \coloneqq x(x-1)\dotsm (x-k+1), \qquad \fallingFactorial{x}{0} \coloneqq 1.\]

The shifted Schur polynomials are defined as the quotient \[\schurShifted_\mu(x_1,\dotsc,x_n) \coloneqq \frac{ \det( \fallingFactorial{x_i+n-i}{\mu_j+n-j} ) } { \det( \fallingFactorial{x_i+n-i}{n-j} ) }.\]

#Properties

The shifted Schur polynomials are stable in the sense that \[\schurShifted_\mu(x_1,\dotsc,x_n) = \schurShifted_\mu(x_1,\dotsc,x_n,0).\]

Let \(H(\mu)\) be the product of hook lengths in the diagram \(\mu.\) The shifted Schur functions \(\schurShifted_\mu(\xvec)\) are the unique family of shifted symmetric functions with degree \(\leq |\mu|\) such that \[\schurShifted_\mu(\lambda) = \delta_{\lambda\mu} H(\mu)\] for all \(\lambda\) such that \(|\lambda| \leq |\mu|.\) This is commonly referred to as the vanishing property.

Let \(l=|\lambda|\) and \(k=|\mu|.\) Then \[\frac{f^{\lambda/\mu}}{f^{\lambda}} = \frac{\schurShifted_\mu(\lambda)}{\fallingFactorial{l}{k}}\] where \(f^{\lambda/\mu}\) is the number of skew standard Young tableaux of shape \(\lambda/\mu.\)

#Combinatorial formula

There is a combinatorial formula, see [OO96], \[\schurShifted_\mu(\xvec) = \sum_{T \in \mathrm{RSSYT}(\mu)} \prod_{s \in \mu} (x_{T(s)} - c(s))\] where \(\mathrm{RSSYT}(\mu)\) is the set of reverse semistandard Young tableaux. These are fillings of \(\mu\) with weakly decreasing rows and strictly decreasing columns. Here, \(c(s)\) is the content of the square \(s.\)

Note that this formula implies that \[\schurShifted_\mu(\xvec) = \schurS_\mu(\xvec) + \text{ lower order terms}.\]

#Jacobi–Trudi identities

Define the shifted elementary and complete homogeneous symmetric functions as \[\elementaryE^*_r(\xvec) = \schurShifted_{(1^r)}(\xvec)\qquad \completeH^*_r(\xvec) = \schurShifted_{(r)}(\xvec).\]

Let \(H^*(u)\) be the formal power series in \(u^{-1}:\) \[H^*(u) \coloneqq \sum_{r\geq 0} \frac{ \completeH^*_r(\xvec) }{ \fallingFactorial{u}{r} } = \prod_{i=1}^\infty \frac{u+i}{u+i-x_i}.\]

Define the automorphism \(\phi\) on the algebra of symmetric functions by \(\phi H^*(u) = H^*(u-1).\)

In [OO96] it is then proved that \[\schurShifted_{\mu}(\xvec) = \det[ \phi^{j-1} \completeH^*_{\mu_i - i +j} ]_{1\leq i,j \leq l} \qquad \schurShifted_{\mu}(\xvec) = \det[ \phi^{1-j} \elementaryE^*_{\mu'_i - i +j} ]_{1\leq i,j \leq m}\] where \(l \geq \length(\mu)\) and \(m \geq \mu_1.\)

#Littlewood–Richardson rule

Since the factorial Schur polynomials have a Littlewood–Richardson rule [MS99], one can find one for the shifted Schur polynomials as well. For examples of a combinatorial interpretation, see [Mol09].

Theorem

The shifted Schur Littlewood–Richardson coefficients are defined via \[\schurShifted_{\lambda}\schurShifted_{\mu} = \sum_{\nu} c^{\nu}_{\lambda \mu} \schurShifted_{\nu}.\] Let \(\mu, \nu \subseteq \lambda.\) Then \[c^{\lambda}_{\mu\nu} = \frac{1}{|\lambda|-|\nu|}\left( \sum_{\nu \to \nu^+} c^{\lambda}_{\mu \nu^+} - \sum_{\lambda^- \to \lambda } c^{\lambda^-}_{\mu \nu} \right)\] where the first sum is taken over all possible ways to add one box to the diagram \(\nu,\) and the second sum is over all ways to remove one box from \(\lambda.\)

This together with the identity \(c^{\lambda}_{\mu \lambda} = \schurShifted_\mu(\lambda)\) which can be computed via the Jacobi–Trudi identity, gives a recursive method to compute the \(c^{\lambda}_{\mu\nu}.\) This recursion has an analogue for the shifted Jack functions.

Note that if \(|\nu| = |\lambda|+|\mu|,\) then \(c^{\nu}_{\lambda\mu}\) is the classical Littlewood–Richardson coefficient for the Schur functions.

#Factorial Schur polynomials

The factorial Schur functions \(\schurFactorial_\lambda(\xvec)\) are a family of non-homogeneous symmetric polynomials, indexed by partitions. They were introduced by L. Biedenharn and J. Louck in [BL89]. The terminology in the literature is not completely uniform: the Biedenharn–Louck inhomogeneous tableau polynomials specialize to shifted Schur functions, while Molev–Sagan use the name factorial Schur functions for a broader factorial/double Schur framework [MS99]. On this site, double Schur polynomials are treated as the Grassmannian subfamily of double Schubert polynomials; this is where the containment relation for factorial Schur polynomials is recorded. The top-degree component is the classical Schur polynomial \(\schurS_\lambda.\) They form a \(\setZ\)-basis for the space of symmetric polynomials (in \(n\) variables).

The factorial Schur functions can be defined using Gelfand–Tsetlin patterns. We sum over all GT-patterns with \(n\) rows and top row \(\lambda.\) This set is in bijection with semistandard Young tableaux of shape \(\lambda\) and maximal entry at most \(n:\) \[\schurFactorial_\lambda(x_1,\dotsc,x_n) \coloneqq \sum_{(m_{ij}) \in GT(\lambda)} \prod_{j=1}^n \prod_{i=1}^j \left( x_j - m_{i,j-1} + i-j \right)_{m_{i,j}-m_{i,j-1}}\] where we are using the falling factorial notation \((a)_k = a(a-1)\dotsm (a-k+1).\) Under the GT-pattern to SSYT bijection, the quantity \(m_{i,j}-m_{i,j-1}\) is sent to the number of times \(j\) appears in row \(i\) (in the SSYT).

From this definition, we can see that \[\schurFactorial_\lambda(x_1,\dotsc,x_n) = \sum_{\mu \downarrow \lambda} \prod_{i=1}^n (x_n - \mu_i - n +i )_{\lambda_i - \mu_i} \cdot \schurFactorial_\mu(x_1,\dotsc,x_{n-1}),\] where we write \(\mu \downarrow \lambda\) for when \[\lambda_1 \geq \mu_1 \geq \lambda_2 \geq \mu_2 \geq \dotsb \geq \mu_{n-1} \geq \lambda_n.\] Equivalently, \(\mu\) can be the row immediately below \(\lambda\) in a GT-pattern.

The factorial Schur polynomials are special cases of the double Schubert polynomials, indexed by Grassmannian permutations.

D. Bump, A. Hardt, and T. Scrimshaw give an algebraic boson–fermion Fock-space construction for Molev’s double supersymmetric Schur functions [BHS25]. By specializing to finitely many variables and shifting parameters, their construction recovers factorial Schur functions.

skew factorial Schur functions can similarly be defined using sums over skew GT-patterns instead (see [CL93]): \[\schurFactorial_{\lambda/\mu}(x_1,\dotsc,x_n) \coloneqq \sum_{(m_{ij}) \in GT(\lambda/\mu)} \prod_{j=1}^n \prod_{i=1}^j \left( x_j - m_{i,j-1} + i-j \right)_{m_{i,j}-m_{i,j-1}}.\] Here, we sum over GT-patterns with \(n+1\) rows, with top row \(\lambda\) and bottom row \(\mu.\)

#Alternant quotient formula

We have \[\schurFactorial_{\lambda}(x_1,\dotsc,x_n) = \frac{1}{\Delta(\xvec)} \sum_{\mu} K_{\lambda \mu} \Delta(\xvec-\mu) \prod_{i=1}^n (x_i)_{\mu_i}\] where \(\Delta(\xvec)\) is the Vandermonde determinant; see [Thm. 5, BL90].

Macdonald proved that the above formula leads to the simpler \[\schurFactorial_{\lambda}(x_1,\dotsc,x_n) = \frac{\left| (x_i)_{\lambda_j+n-i} \right| } { \prod_{i \lt j} (x_i - x_j) },\] which is analogous to the alternant quotient formula for Schur functions, see [Thm. 3.2, CL93] for a proof.

#Jacobi–Trudi

There is a Jacobi–Trudi identity for skew factorial Schur functions; see [CL93].

D. Heilman studies combinatorial transition matrices between ordinary Schur functions and factorial Schur functions in his dissertation [Hei13]. Starting from the Molev–Sagan expansion of factorial Schur functions into Schur functions, he gives a combinatorial description of the reverse change of basis using new objects and a sign-changing involution.

#Generalized factorial Schur functions

Molev–Sagan (see [MS99]) consider the generalized factorial Schur functions defined via the alternant formula \[\schurFactorial_{\lambda}(x_1,\dotsc,x_n | \avec) = \frac{\left| (x_i|a)_{\lambda_j+n-i} \right|} { \prod_{i \lt j} (x_i - x_j) },\] where \((y|a)_k \coloneqq (y-a_1)(y-a_2)\dotsm (y-a_k).\) Thus, the classical factorial Schur functions are a specialization of the Molev–Sagan family: \[\schurFactorial_{\lambda}(x_1,\dotsc,x_n ) = \schurFactorial_{\lambda}(x_1,\dotsc,x_n | 0,1,2,3,\dotsc).\]

These also admit a tableau definition: \[\schurFactorial_{\lambda}(x_1,\dotsc,x_n | \avec) = \sum_{T \in \SSYT(\lambda)} \prod_{\alpha \in \lambda} \left( x_{T(\alpha)} - a_{T(\alpha)+c(\alpha)} \right)\] where \(c(\alpha)\) is the content of the box \(\alpha.\) These are essentially the same as the double Schur functions; see [Mol09].

#Littlewood–Richardson rule

The factorial Schur functions admit a Littlewood–Richardson rule; see [MS99].

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