Root-sensitive k-Schur functions and an exponent-profile obstruction

Definitions and mathematical statement

The source asks to generalize $k$-Schur functions beyond root-system type A. It does not specify which properties must be retained: a symmetric-function realization, affine-Schubert basis, Hopf structure, positivity, or a Macdonald-type parameter deformation. These are distinct versions of the request.

The supplied aid reports an established geometric specialization at deformation parameter $t=1$ for every crystallographic root system: the affine-Grassmannian Schubert basis, equivalently its triangular Peterson/Fomin--Stanley realization. For a simply connected semisimple complex group $G$, its homology $H_*(\operatorname{Gr}_G;{\mathbb{Z}})$ has a Schubert basis indexed by affine Grassmannian elements of the affine Weyl group. In type $A_r$, $G=\mathrm{SL}_{r+1}$, this recovers the $k=r$ Schur basis at $t=1$ in \[ \Lambda_{(r)}={\mathbb{Z}}[h_1,\ldots,h_r],\qquad\deg h_i=i. \] Use half the topological homological grading throughout. This is not a supplied solution for noncrystallographic root systems or every parameter-dependent $k$-Schur theory. The aid reports exact symmetric-function models in types C and B, while a uniform Macdonald-type deformation remains incomplete in that attempt.

Root-degree obstruction reported in the aid. Let the root system have rank $r$ and exponents $m_1,\ldots,m_r$. Rationally, the affine-Grassmannian-compatible homology algebra has polynomial generator degrees $m_i$, hence \[ \operatorname{Hilb}(H_*(\operatorname{Gr}_G;{\mathbb{Q}});u) =\prod_{i=1}^r(1-u^{m_i})^{-1}. \] Equivalently, its rational augmentation indecomposables have precisely those degrees, with multiplicities. If a proposed generalization used the unchanged graded type A algebra ${\mathbb{Q}}[h_1,\ldots,h_r]$ with degrees $1,\ldots,r$, equality of graded Hilbert series would require \[ \{m_1,\ldots,m_r\}=\{1,\ldots,r\}. \] For finite crystallographic root systems this forces type $A_r$, up to the usual low-rank isomorphisms. Reducible systems have one exponent 1 per component and must retain their multiplicities. The obstruction is graded and root-sensitive; it is not a claim that abstract ungraded polynomial algebras in $r$ variables are nonisomorphic.

The aid's integral type $C_2$ witness uses one-row Schur $P$-functions. Normalize them by \[ \prod_{a\ge1}\frac{1+x_a u}{1-x_a u} =1+2\sum_{j\ge1}P_j u^j. \] In the stated type C model the ring is \[ {\mathbb{Z}}[P_1,P_3],\qquad\deg P_1=1,\quad\deg P_3=3, \] with \[ P_2=P_1^2,\qquad P_4=2P_1P_3-P_1^4. \] Thus $P_2$ cannot serve as an independent degree-two generator, and the correct exponent profile is $(1,3)$. These relations provide an integral illustration, not a uniform construction for all other types or a Macdonald deformation.

The remaining formulation task is to specify which representation-theoretic and combinatorial properties define the desired non-A functions and how a parameter $k$ and deformation variables should vary. The geometric $t=1$ basis and exponent constraint answer parts of that request without selecting a unique general theory.

Short English statement

The aid reports affine-Grassmannian Schubert homology as the crystallographic $t=1$ generalization. Its generator degrees must follow root exponents, so the unchanged graded type A algebra works only in type A. A type $C_2$ Schur-$P$ example shows the required degree change, while a uniform parameter deformation remains outside the supplied result.

Sources

• Local UnsolvedMath registry, permanent ID 20002901, catalogue entry; canonical aim-special-functions-notes.json, notes[5].
• AIM Generalized Kostka polynomials, Problem 6, https://aimath.org/WWN/kostka/prob2.pdf. No individual proposer or authoritative bibliography is supplied.
• The aid names Peterson/Fomin--Stanley realizations and type B/C symmetric-function models without full titles, authors for those models, dates or theorem references.
• Machine-generated research attempt 1, partial-result/partially-solved labels, reported rigor proved and low novelty confidence. Its exponent-profile obstruction and integral type $C_2$ illustration are not peer reviewed and require independent verification.