Webs and smooth components of two column Springer fibers

WEBS AND SMOOTH COMPONENTS OF TWO COLUMN SPRINGER FIBERS MIKE CUMMINGS A B S T R A C T . W ebs and Springer fibers are separately important objects in representation theory: webs give a diagrammatic calculus for tensor invariants of sl k , and the cohomology group of Springer fibers can be used to construct the irreducible representations of the symmetric group. Fung’s 1997 thesis gave the first evidence of a connection between sl 2 webs and Springer fibers, showing that webs naturally index and describe the components of certain “two row” Springer fibers. However , this case is known to be far from generic. This paper deepens this connection with a similar correspondence in the substantially more complicated “two column” case. In particular , and building on works of Fresse, Melnikov , and Sakas-Obeid, we use webs to give a clean characterization of the smooth components of two column rectangle Springer fibers and a simple description of the geometry of these smooth com- ponents. W e also show that the P oincaré polynomial of the smooth components is invariant under the natural dihedral action on the corresponding webs. 1. I N T R O D U C T I O N The set of nilpotent elements in a semisimple Lie algebra g forms a normal variety [ Kos63b ] that relates to the geometry of Schubert varieties [ Kos61 ; K os63a ; Car93 ] , and is of significant current interest, e.g., [ MV22 ; Nev24 ; CHY25 ] . Springer [ Spr69 ] constructed a resolution of this variety , whose fibers are now known as Springer fibers . In this paper , we consider the type A case g = sl k . Here, Springer fibers are subvarieties of the complete flag variety SL k ( C ) / B , where B is the Borel subgroup of upper-triangular matrices, and are indexed by integer partitions η = ( η 1 , η 2 , . . . ) of n . In this case, Springer [ Spr76 ] (see also [ Slo80 ; Lus81 ; BM83 ; PR21 ] ) showed that the cohomology of the Springer fiber S η carries an action of the symmetric group S n and, moreover , the top-graded piece of the cohomology is isomorphic to the Specht module S η . In other contexts, Springer fibers appear in the construction of convolution algebras [ SW12 ] , in knot homology [ Kho02 ; SS06 ; CK08 ] (where the cohomology of two row rectangular Springer fibers relates to categorification of the Jones polynomial [ Jon85 ] ), and in the combinatorics of Y oung tableaux (for instance, [ Spa76 ; vLee00 ; FMSO15 ] ). Despite this long and rich history , the geometry and topology of Springer fibers are poorly understood in general. Outside of some special cases, we only know that they are connected [ Spa82 , Chapter II.1 ] and the following. Theorem 1.1 ( [ Spa76 ] (see also [ V ar79 ; vLee00 ] )) . The Springer fiber S η is equidimensional of dimension P i ≥ 1 ( i − 1 ) η i . Moreover , the components of S η are in bijective correspondence with the standard Y oung tableaux of shape η . The present manuscript treats the two column case, where η = ( 2, . . . , 2, 1, . . . , 1 ) . In this case, Fresse and Melnikov [ FM11 ] characterize the tableaux that correspond to smooth com- ponents of S η and, with Sakas-Obeid [ FMSO15 ] , describe the geometry of each smooth compo- nent from its tableau. Our main result is a diagrammatic reinterpretation of these two column results, using degree two sl k webs . Date : F ebruary 20, 2026. 1 2 1 2 3 9 4 1 0 5 1 2 6 1 3 7 1 4 8 1 5 1 1 1 6 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 6 2 8 3 9 4 1 0 5 1 3 7 1 4 1 1 1 5 1 2 1 6 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 F I G U R E 1. T wo column tableaux and their associated degree two sl 8 webs. W ebs are planar graphs embedded in a disc and themselves have a rich history , originating with Kuperberg in 1996 [ Kup96 ] as diagrammatic tools for computation in the representations of Lie groups and their quantum deformations. W ebs have important applications to areas such as quantum topology [ Kho04b ; Hig23 ; Bod22 ; BW25 ] , cluster algebras [ FP16 ; FP23 ] , dimer models [ Lam15 ; FLL19 ; DKS24 ] , dynamical algebraic combinatorics [ PPR09 ; HR22 ] , the geometric Satake correspondence [ FKK13 ] , and skein algebras [ LS24 ; IK25 ; SSW25 ] . W ebs are well-understood in the sl 2 case, where the T emperley –Lieb basis webs are noncrossing matchings , e.g., [ RTW32 ; TL71 ; KR84 ] . Our main result is that the degree two sl k webs of [ Fra23 ; GPPSS25 ] characterize and describe the geometry of the smooth components of two column rectangle Springer fibers S ( 2,2,...,2 ) . Main Theorem. Let W be the degree two sl k web associated to a component S W of the two column rectangle Springer fiber S ( 2,2,...,2 ) . • (Theorem 3.2 ) S W is smooth if and only if the underlying graph of W is a forest. • (Theorem 3.8 ) Suppose that S W is smooth and W has 2 k boundary vertices. If the underlying graph of W is disconnected, then let i be the maximum integer such that the vertices 1, 2, . . . , i are in the same claw of W . Then, the component S W is an iterated fiber bundle with base  Fl ( i ) × Fl ( k ) , P i , P i + 1 , . . . , P k − 1  . Otherwise, the interior of W is connected. Denote by i , j , m the number of vertices in the first, second, and third claw , respectively , and let ℓ be the multiplicity of the edge between the second claw and the filled internal vertex. Then, the component S W is an iterated fiber bundle with base  Fl ( i ) × Fl ( j ) , Gr ℓ ( m ) , P k − m , P k − m + 1 , . . . , P k − 1  . W e demonstrate this theorem in Examples 1.3 , 3.3 and 3.9 . Our results are reinterpretations of works of Fresse, Melnikov , and Sakas-Obeid [ FM11 ; FMSO15 ] that were given in terms of standard Y oung tableaux. Example 1.3 contrasts our two rules. This perspective led to a correction [ CM26 ] of an enumeration of Mansour [ Man25 ] . Experts have known since Fung’s thesis [ Fun97 ; Fun03 ] that webs govern the geometry and topology of two row rectangle Springer fibers S ( k , k ) . Indeed, the noncrossing matching diagrams in work of Fung and others [ R us11 ; SW12 ] , coincide with sl 2 webs. However the two row case is far from generic; for instance, each component is a smooth, iterated P 1 -bundle. The 3 same does not hold for arbitrary Springer fibers, which admit singular components [ FM10 ] . Although not smooth, the components of the two column Springer fibers, which we consider in this paper , are normal, Cohen–Macaulay , and have rational singularities [ PS12 ] . Since webs describe the geometry of two row and smooth components of two column Springer fibers, we record the following question, which has been asked by experts (e.g., [ T ym25 ] ) but has not yet appeared explicitly in the literature. Question 1.2. Do webs describe the geometry of the components of Springer fibers in general? From their embedding into a disc, webs inherit a natural dihedral action, corresponding to rotation and reflection of the underlying graph. Combinatorially , rotation of webs corresponds to dynamics called promotion on the associated tableaux [ Whi07 ; PPR09 ; GPPSS24 ] . Also, webs correspond to vectors in certain invariant spaces of tensors of representations. It is de- sirable to determine a rotation-invariant web basis for this space: a set of webs that is closed under diagrammatic rotation (up to sign) and that corresponds to a basis of the invariant space. Kuperberg [ K up96 ] describes such a basis in the sl 2 and sl 3 cases, yet it took nearly 30 years to solve the degree two sl k [ GPPSS25 ] and sl 4 [ GPPSS26 ] cases. T o complement the combinato- rial and representation-theoretic interpretations of rotation, we give the following geometric interpretation. Corollary (Theorem 3.10 ) . T wo smooth components of S ( 2,2,...,2 ) have the same P oincaré polyno- mial if and only if their webs are a rotation or reflection of one another . Example 1.3. Consider the degree two sl 8 webs and their corresponding tableaux given in Figure 1 . The first web is a tree, hence corresponds to a smooth component of its associated Springer fiber (Theorem 3.2 ). (The disc in which the web is embedded does not contribute any edges to the graph.) Its first claw involves the boundary vertices 2, 3, . . . , 8. Hence, from Theorem 3.8 , the component corresponding to the first web is an iterated fiber bundle whose base is given by  Fl ( 7 ) × Fl ( 3 ) , Gr 5 ( 6 ) , P 2 , P 3 , . . . , P 7  . The second web contains a cycle, drawn in red, hence corresponds to a singular component of the Springer fiber . W e contrast this with the procedure outlined in [ FM11 ; FMSO15 ] , which we discuss in Section 2.3 . First, to check for singularities, we note that both tableaux have exactly three entries j in their first columns such that j + 1 is in the second column. In the first tableau, when i = 1, the i -th entry in the second column is exactly 2 i , so the corresponding component is smooth. However , for the second tableau, this does not occur for any i < k , where k = 8 is the number of rows; hence this tableau corresponds to a singular component. T o describe the geometry of the component corresponding to the first tableau, we set a = 5, b = 2, and c = 1 per Definition 2.4 , and use Theorem 2.5 to conclude that the corresponding base is  Fl ( a + b ) × Fl ( b + c ) , Gr a ( a + c ) , P b , P b + 1 , . . . , P k − 1  , which agrees with our first calculation. Outline of the paper . W e first discuss the background and relevant results from the litera- ture of Springer fibers and webs in Section 2 . Section 3 treats the case of two row rectangu- lar Springer fibers, each subsection treating one of the three main results above: character- ization of smooth components, description of the geometry of the smooth components, and the P oincaré polynomials. W e discuss in Appendix A an extension of our results to the non- rectangular two column case using noncrossing matching and ray diagrams . 4 A C K N O W L E D G E M E N T S The author thanks his advisor , Oliver P echenik, for invaluable suggestions and feedback that substantially improved this manuscript. W e thank R onit Mansour for helpful discussions about [ Man25 ] . W e are also grateful to Stephan Pfannerer and Joshua P . Swanson for helpful conversations about webs and for Sage code used to produce many of the figures. W e thank Joel Kamnitzer , Jake Levinson, and Martha Precup for helpful conversations—particularly those involving R emark 3.11 —and Steven N. Karp for his comments on an earlier draft. Lastly , we acknowledge [ OEIS ] for the discovery of the equinumerosity in Corollary 3.5 . The author is supported by NSERC Alexander Graham Bell CGS-D 588999-2024, a University of W aterloo President’s Graduate Scholarship, and Oliver P echenik’s NSERC Discovery Grant RGPIN-2021-02391. 2. B A C K G R O U N D In this section we review standard definitions and results from the literature relevant to the present manuscript. First, we establish the necessary notation for integer partitions and tableaux in Section 2.1 . Then, in Section 2.2 , we discuss Springer fibers and the correspon- dence with tableaux, and specialize in Section 2.3 to the two column case. Section 2.4 discusses webs, and we again specialize to the two column case in Section 2.5 . 2.1. Combinatorics of tableaux. W e briefly review some combinatorial notions, referring the reader to [ Sag20 ] for further details. An integer partition of n ∈ Z + (or simply , a partition of n ) is a weakly decreasing sequence η = ( η 1 , η 2 , . . . ) of positive integers such that P i η i = n . The Y oung diagram D η of η = ( η 1 , η 2 , . . . ) is a left-justified grid of n boxes such that the i -th row has exactly η i -many boxes. W e draw our diagrams in English notation , so they are indexed using matrix coordinates and the first row is at the top of the diagram. The conjugate η ∗ of η is the partition obtained by taking the transpose of the diagram D η , that is, swapping its rows and columns. W e often describe integer partitions by the shape of their diagram. F or instance, the diagram of the partition η = ( η 1 , η 2 ) has two rows, so η is a two row partition and η ∗ is a two column partition, and when η 1 = η 2 , we say that η ∗ is a two column rectangle . 1 3 2 4 5 7 6 8 1 3 2 4 5 6 7 8 1 2 3 4 5 6 7 8 F I G U R E 2. The diagram of the two column rectangle η = ( 2, 2, 2, 2 ) = ( 4, 4 ) ∗ and three standard Y oung tableaux of shape η . F or any partition η of n , a filling of its diagram D η is a bijective assignment of the values [ n ] : = { 1, 2, . . . , n } to the boxes in D η . A standard Y oung tableau of shape η is a filling of D η that is increasing along rows and columns from left-to-right and top-to-bottom, respectively . W e denote by SYT ( η ) the set of all standard Y oung tableaux of shape η . The number of standard Y oung tableaux of a given shape is given by the hook-length formula [ Sag20 , Theorem 7.3.1 ] . Figure 2 illustrates the diagram D η and three standard Y oung tableaux of shape η = ( 2, 2, 2, 2 ) . 5 2.2. Springer fibers. Springer fibers were introduced in work of Springer [ Spr69 ; Spr76 ] , arising from his study of the nilpotent cone , the variety of nilpotent elements in a semisimple Lie algebra. F or an overview of the geometry and combinatorics of Springer fibers, we direct the reader to the survey paper of T ymoczko [ T ym17 ] and references therein. The Jordan canonical form of a nilpotent operator on C n corresponds uniquely to a partition of n , so we abuse notation and write η both for a partition and its corresponding nilpotent operator . W e work in type A , where the (complete) flag variety Fl ( n ) in C n is the set of nested C -vector subspaces V • = ( V 0 ⊂ V 1 ⊂ · · · ⊂ V n ) such that dim C V i = i for all i . Each element V • of Fl ( n ) is called a flag . The Grassmannian Gr d ( n ) is the set of d -dimensional vector subspaces of C n . Projective space P n is the set of lines in C n + 1 , so P n = Gr 1 ( n + 1 ) and dim C P n = n . The Springer fiber S η associated to a nilpotent operator η : C n → C n is the subvariety of Fl ( n ) consisting of the flags V • satisfying η · V i ⊆ V i − 1 for all i . F or any invertible g ∈ S L n ( C ) , there is an isomorphism S η ∼ = S g η g − 1 , so we freely take η to be in Jordan form. This gives a bijective correspondence between partitions of n and Springer fibers in Fl ( n ) . W e describe the Springer fiber S η with the same adjectives associated to η , for instance, if η = ( η 1 , η 2 ) ∗ is a two column partition, then we say that S η is a two column Springer fiber . T o conclude this subsection, we survey some relevant results on the geometry of Springer fibers. In general, the Springer fiber S η is connected [ Spa82 , Chapter II.1 ] and its components are singular [ V ar79 ; FM10 ] . Moreover , S η is equidimensional of dimension P i ≥ 1 ( i − 1 ) η i and there is a bijective correspondence between its irreducible components and the standard Y oung tableaux of shape η ; see Theorem 1.1 . F or T ∈ SYT ( η ) , we write S T for the corresponding irreducible component of S η . The dimension formula for S η is obtained from the diagram of η by placing the value i − 1 in every box in the i -th row of D η , and summing the values in all the boxes. F or instance, when η = ( 3, 2, 1 ) , we obtain 0 0 0 1 1 2 , so S ( 3,2,1 ) has dimension 4. When η = ( η 1 , 1, . . . , 1 ) is a hook shape , Fung [ Fun03 ] describes the topology of the compo- nents of S η and their pairwise intersections. Fung [ Fun03 ] and [ CK08 ; W eh09 ; R T11 ; R us11 ; SW12 ; IL W22 ] describe the geometry and topology of components of two row Springer fibers and the pairwise intersections of these components. Geometrically , in this two row case, the components are smooth, iterated P 1 -bundles, and the number of components in this bundle is described by certain noncrossing matching and ray diagrams , which are in correspondence with two row standard Y oung tableaux. Karp and Precup [ KP25 ] characterize the components of a Springer fiber that are equal to a Richardson variety . Springer fibers are paved by affines [ Spa76 ] and, in the two row case, this paving is described explicitly by noncrossing matching diagrams [ GNST25 ] . Every component of S η is smooth only when η is either a hook [ V ar79 ] , has two rows [ Fun03 ] , or is of the form η = ( η 1 , η 2 , 1 ) or η = ( 2, 2, 2 ) [ Fre09 ; FM10 ] . When η = ( η 1 , η 2 ) ∗ is a two column partition, the components of S η are normal, Cohen–Macaulay , and have rational singularities [ PS12 ] . 2.3. Geometry for two column rectangle Springer fibers. W e henceforth consider two col- umn rectangle Springer fibers S η , that is, where η = ( k , k ) ∗ for some k . In Appendix A , we discuss extensions of our work to the arbitrary two column case, using noncrossing matching and ray diagrams . Although these diagrams do correspond to two column standard Y oung tableaux, they are not webs in the representation-theoretic sense. 6 F or any two column Springer fiber S η , Fresse and Melnikov [ FM11 ] characterize the stan- dard Y oung tableaux corresponding (under Theorem 1.1 ) to smooth components of S η . Then, in their work with Sakas-Obeid [ FMSO15 ] , they describe the geometry of these smooth com- ponents. W e now summarize their results in the two column rectangle case. Fix a two column rectangle partition η = ( k , k ) ∗ and standard Y oung tableau T ∈ SYT ( η ) . Denote by col i ( T ) the set of entries in the i -th column of T . W e write col 1 ( T ) = { a 1 < a 2 < · · · < a k } and col 2 ( T ) = { b 1 < b 2 < · · · < b k } . Define τ ∗ ( T ) = { j ∈ col 1 ( T ) | j + 1 ∈ col 2 ( T ) } to be the set of entries j in the first column of T for which j + 1 is in the second column. From the set τ ∗ ( T ) , we have the following characterization of the smooth components of S η . R ecall that for a tableau T ∈ SYT ( η ) , we write S T for its corresponding component in S η under the bijection in Theorem 1.1 . The following also appears as [ FMSO15 , Theorem 2 ] . Theorem 2.1 ( [ FM11 , Theorem 1.2 ] ) . Let η = ( k , k ) ∗ and T ∈ SYT ( η ) . Then, the corresponding irreducible component S T of S η is smooth if and only if | τ ∗ ( T ) | ≤ 3 and, when | τ ∗ ( T ) | = 3 , there is some i ∈ [ k − 1 ] such that b i = 2 i . Consider the standard Y oung tableaux in Figure 2 , say T 1 , T 2 , and T 3 from left-to-right. These tableaux correspond to irreducible components of the Springer fiber S ( 2,2,2,2 ) . W e leave it to the reader to verify that | τ ∗ ( T 1 ) | = 2 and | τ ∗ ( T 2 ) | = | τ ∗ ( T 3 ) | = 3 and, moreover , that S T 1 and S T 2 are smooth components of S η but that S T 3 is singular . W e denote by SYT smooth ( η ) the set of tableaux T in SYT ( η ) whose corresponding component S T is smooth. The geometry of a smooth component of a two column Springer fiber is an iterated fiber bundle . The following definition is as in [ FMSO15 , §2.3 ] . Definition 2.2. Let X and B 1 , . . . , B m be algebraic varieties. W e say that X is an iterated fiber bundle of base ( B 1 ) if X and B 1 are isomorphic. W e say that X is an iterated fiber bundle of base ( B 1 , . . . , B m ) if there is a fiber bundle X → B m whose fiber is an iterated fiber bundle of base ( B 1 , . . . , B m − 1 ) . F or two examples, B 1 × · · · × B m is an iterated fiber bundle with base ( B 1 , . . . , B m ) , and the flag variety Fl ( n ) is an iterated fiber bundle with base ( P 1 , P 2 . . . , P n − 1 ) . R emark 2.3. Notice that if in Definition 2.2 some B i is a point, it follows that X is isomorphic to an iterated fiber bundle with base ( B 1 , . . . , ˆ B i , . . . , B m ) . Consequently , we may freely omit any component of the base that is a point. In [ FMSO15 ] , the geometry of smooth components of two column Springer fibers is de- scribed using the following data from the corresponding standard Y oung tableau. Definition 2.4 ( [ FMSO15 , §2.4 ] ) . Let T ∈ SYT smooth (( k , k ) ∗ ) . • If τ ∗ ( T ) = { α } , then a T = α − k , b T = k , and c T = k − α , • If τ ∗ ( T ) = { α < β } , then a T = k − ( β − α ) , b T = β − k , and c T = k − α , • If τ ∗ ( T ) = { α < β < γ } , then a T = k − ( γ − β ) , b T = ( γ − α ) − k , and c T = k − ( β − α ) . The values a T , b T , c T are nonnegative and satisfy b T ≥ 1 and a T + b T + c T = k [ FMSO15 , §2.4 ] . 7 Theorem 2.5 ( [ FMSO15 , Theorem 3 ] ) . Let T ∈ SYT smooth (( k , k ) ∗ ) and define a T , b T , and c T as in Definition 2.4 . Then, S T is an iterated fiber bundle with base  Fl ( a T + b T ) × Fl ( b T + c T ) , Gr a T ( a T + c T ) , P b T , P b T + 1 , . . . , P k − 1  . Their proof [ FMSO15 , Section 3 ] uses combinatorics of tableaux— jeu de taquin and a dele- tion procedure they call projection —to reduce to the case where the geometry can be under- stood from a description of the component similar to that of [ V ar79 , Proposition 2.2 ] . W e will see, in Theorem 3.8 , a reinterpretation of this result in terms of webs . As an application of our work, we consider the P oincaré polynomials of the smooth compo- nents of two column rectangle Springer fibers. F or an algebraic variety X , denote by H i ( X , C ) the i -th graded piece of its sheaf cohomology group. The P oincaré polynomial of X is P X ( q ) = X i ≥ 0 q i dim C H i ( X , C ) . Since H i ( X , C ) = 0 for all i > dim X , only finitely-many terms of this sum are nonzero. If, moreover , the variety X is an iterated fiber bundle with base ( B 1 , B 2 , . . . , B ℓ ) , then P X ( q ) = P B 1 ( q ) P B 2 ( q ) · · · P B ℓ ( q ) . In view of Theorem 2.5 , the P oincaré polynomials of the components of our Springer fibers are products of the P oincaré polynomials of projective spaces, flag varieties, and Grassmanni- ans. These P oincaré polynomials are given in terms of q -integers . F or any n ∈ N , define the q -integer [ n ] q = 1 + q + q 2 + · · · + q n − 1 . Notice that when q = 1 we have [ n ] q = 1 = n . Using this, we define q -factorials and q -binomials in analogy with their usual definitions. That is, we take [ n ] q ! = [ n ] q [ n − 1 ] q · · · [ 2 ] q [ 1 ] q and, for any integer d ≤ n ,  n d  q = [ n ] q ! [ d ] q ! [ n − d ] q ! . When d > n , the q -binomial is 0. If d = 0, then the q -binomial is 1 if and only if n = 0, and 0 otherwise. Arguing inductively and using a q -analogue of the binomial identity  n d  =  n − 1 d  +  n − 1 d − 1  , one can show that the q -binomial is indeed a polynomial in q [ Sag20 , Theorem 3.2.3 ] . W e henceforth suppress the subscript q when there is no ambiguity . F act 2.6. The P oincaré polynomials of P n , Fl ( n ) , and Gr d ( n ) are, respectively , P P n ( q ) = [ n + 1 ] , P Fl ( n ) ( q ) = [ n ] !, and P Gr d ( n ) ( q ) =  n d  . The P oincaré polynomials for P n and Fl ( n ) follow from that of Gr d ( n ) . Indeed, since P n = Gr n ( n + 1 ) , we have P P n ( q ) = h n + 1 1 i = [ n + 1 ] . Also, since Fl ( n ) is an iterated fiber bundle with base ( P 1 , P 2 , . . . , P n − 1 ) , we have P Fl ( n ) ( q ) = P P ( q ) P P 2 ( q ) · · · P P n − 1 ( q ) = [ n ] !. Theorem 2.7 ( [ FMSO15 , Theorem 4 ] ) . Let η be a partition and T , T ′ ∈ SYT smooth ( η ) be tableaux corresponding to smooth components of S η . Define ( a T , b T , c T ) and ( a T ′ , b T ′ , c T ′ ) as in Defini- tion 2.4 for T and T ′ , respectively . Then, P S T ( q ) = P S T ′ ( q ) if and only if either • 0 ∈ { a T , c T } and 0 ∈ { a T ′ , c T ′ } , or , • 0 / ∈ { a T , a T ′ , c T , c T ′ } and { a T , b T , c T } = { a T ′ , b T ′ , c T ′ } . In Theorem 3.10 , we will see that this condition is equivalent to when the corresponding webs are in the same dihedral orbit. 8 2.4. W ebs. W ebs originated in work of K uperberg [ K up96 ] as a diagrammatic computation tool in the study of invariants of tensor representations of sl 2 , sl 3 , and their quantum deforma- tions. The sl 2 webs coincide with noncrossing matchings (which we will define in Section 2.5 ) and are sometimes referred to as T emperley -Lieb webs . These webs are well-understood, and they can be traced back at least to work of R umer , T eller , and W eyl; see [ R TW32 ; TL71 ; KR84 ] . F or the purpose of diagrammatic computations, it is desirable to obtain rotation-invariant web bases : a set of webs that is closed under diagrammatic rotation (up to sign) and that corre- sponds to a basis in the tensor invariant space. It is an open problem to find rotation-invariant web bases for general sl k ; the sl k degree two [ GPPSS25 ] and sl k [ GPPSS26 ] cases were only recently solved. W ebs in this manuscript follow the k -valent, bicoloured by filled and unfilled vertices, and hourglass conventions of [ GPPSS25 ; GPPSS26 ] . That is, our webs have multiedges which are drawn as hourglasses , so that the clockwise order of edges is the same between any two adjacent vertices. Moreover , they are plabic , as defined by P ostnikov [ P os06 ] , meaning they have a pla nar embedding and bic olouring into filled and unfilled vertices. W e consider webs up to planar isotopy fixing the boundary disc. The vertices are partitioned into boundary and internal vertices. The boundary vertices are filled, labelled 1, 2, . . . , 2 k , and lie on the boundary disc. Although we draw the web with the embedding of underlying graph in a disc, this disc does not contribute any edges to the graph. In particular , the boundary vertices have degree 1. A web is k -valent (or k -regular ) if each internal vertex of the underlying graph has degree k . A claw of a web is the set of boundary vertices that are incident to a common internal vertex. 2.5. Degree two sl k webs. In this paper , we consider the degree two sl k webs of [ Fra23 ; GPPSS25 ] . The degree two prefix arises from the fact that the corresponding tensor invariant space can be identified with a homogeneous piece of the coordinate ring of the Grassmannian spanned by products of pairs of Plücker coordinates with disjoint support. Certain equivalence classes of degree two sl k webs are in bijection with two column rectangular standard Y oung tableaux [ Fra23 , Proposition 2.14 ] , hence correspond to the irreducible components of the Springer fiber of the same partition shape by Theorem 1.1 . Denote by B 2 k the set of degree two sl k webs. Diagrammatically , this is the set of k -valent hourglass plabic graphs with 2 k boundary vertices, as defined in Section 2.4 . W e consider B 2 k up to an equivalence relation called moves , described in [ GPPSS25 ] (c.f ., [ Fra23 , Proposition 2.14 ] ). Our results are well-defined with respect to this equivalence relation, in the sense that any two move-equivalent webs have the same structure needed for our results. See, for instance, Examples 3.3 and 3.7 . W e now describe a bijection from standard Y oung tableaux of shape ( k , k ) ∗ to B 2 k when k ≥ 2. Our presentation follows that of [ GPPSS25 ] , which itself is an adaptation of work of Fraser [ Fra23 ] . Despite the map involving a choice, the result equivalence class in B 2 k is well-defined [ Fra23 , Proposition 2.14 ] . The bijection is the following composition of maps SYT (( k , k ) ∗ ) →  noncrossing (perfect) matchings of 2 k vertices  → k [ s = 2  weighted dissections of an s -gon  → k [ s = 2  weighted triangulations of an s -gon  → B 2 k . 9 Let G = ( V , E ) be a graph with vertices V and edges E ⊆ V × V . A matching of G is a subset of edges M ⊆ E such every vertex in the subgraph ( V , M ) has degree at most 1. When every vertex in ( V , M ) has degree exactly 1, we say that M is a perfect matching . W e embed the vertices 1, 2, . . . , 2 k of ( V , M ) sequentially clockwise around the boundary of a disc, and say that the matching M is noncrossing if the edges can be drawn inside the disc without intersection. As an abuse of terminology , we write noncrossing matching to mean noncrossing perfect matching. W e now describe each of the above maps in turn. The first map is the usual bijection between two column rectangle standard Y oung tableaux and noncrossing matchings. That is, it is the unique map that sends T ∈ SYT (( k , k ) ∗ ) to a noncrossing matching of 2 k vertices such that each edge { i , j } with i < j has i in the first column of T and j in the second column. Concretely , write the entries in the second column of T as col 2 ( T ) = { b 1 < b 2 < · · · < b k } . W e match the edges b 1 , b 2 , . . . , b k in this order , and match b i with the maximum unmatched entry in col 1 ( T ) that is less than b i . Figure 3 gives an example of such a matching built step-by-step. 1 3 2 4 5 6 7 8 9 1 0 1 2 3 4 5 6 7 8 9 1 0 1 2 3 4 5 6 7 8 9 1 0 1 2 3 4 5 6 7 8 9 1 0 1 2 3 4 5 6 7 8 9 1 0 1 2 3 4 5 6 7 8 9 1 0 F I G U R E 3. Step-by-step construction of a noncrossing matching from the corre- sponding two column rectangular tableau. Given a noncrossing matching on 2 k vertices with k ≥ 2, let i 1 < · · · < i s be the subset of boundary vertices { 1, 2, . . . , 2 k } whereby { i j , i j + 1 } mod 2 k is an edge in the matching. Notice that 2 ≤ s ≤ k . Construct an s -gon by identifying the vertices i j + 1, . . . , i j + 1 for all j , working mod 2 k . Denote by V j the corresponding vertex, say , V j = { i j + 1, . . . , i j + 1 } , and draw each V j unfilled. Between V j and V ℓ introduce an edge with weight m , where m is the number of edges between vertices of V j and V ℓ in the noncrossing matching. If m = 0, then there is no edge between V j and V ℓ . F or the example in Figure 3 , we obtain 4 vertices V 1 , V 2 , V 3 , and V 4 , corresponding to { 3, 4, 5 } , { 6, 7 } , { 8, 9 } and { 10, 1, 2 } , respectively . Figure 4 shows the resulting dissection with edge weights. Then, triangulate of the s -gon by including nonintersecting diagonals, represented as weight 0 edges. This involves a choice, but any choice gives a web in the same move-equivalence class in B 2 k ; see our discussion at the beginning of this subsection. The penultimate step takes this weighted triangulation and produces the interior of our desired web. In each of the triangles in the triangulation, place a filled vertex. Add edges from this filled vertex to the three vertices of its bounding triangle. The weight of an edge e in the interior of the web is the sum of weights in the triangulation of the edges opposite e in the following sense: In the triangulation, there is an edge f opposite of e which bisects the s -gon. The desired weight is the sum of the weight of f and the weights of all edges in the triangulation in the bisected component that does not include e . Figure 4 demonstrates this procedure with our running example. W e now obtain our web by replacing the vertices of the s -gon as follows. T ake, in order , the unfilled vertices V j = { i j + 1, . . . , i j + 1 } of the s -gon. Place V j inside the embedding disc, 10 1 1 1 2 1 0 , 1 , 2 3 , 4 , 5 6 , 7 8 , 9 1 1 1 2 0 1 0 , 1 , 2 3 , 4 , 5 6 , 7 8 , 9 1 1 1 2 0 1 0 , 1 , 2 3 , 4 , 5 6 , 7 8 , 9 1 2 2 1 3 1 F I G U R E 4. Left-to-right: The weighted dissection, a weighted triangulation, and the interior of the resulting web (in green ) corresponding to the noncrossing matching in Figure 3 . and introduce filled boundary vertices for each of i j + 1, . . . , i j + 1 . Place these boundary vertices sequentially clockwise on the disc (which, recall, does not contribute any edges to the graph). The interior of the web is exactly as constructed in the previous step. Lastly , replace each internal edge of weight m with a m -hourglass edge; see Figure 5 . An hourglass is a multiedge where the edges are twisted such that the clockwise order of edges is the same between the two endpoint vertices. 1 2 3 4 5 6 7 8 9 1 0 1 2 3 4 5 6 7 8 9 1 0 F I G U R E 5. The example from Figure 3 continued. Left: The web obtained from the triangulation in Figure 4 . Right: The web obtained by choosing the other diagonal in the triangulation. W e make the following notational convention. Definition 2.8. Let η = ( k , k ) ∗ and T ∈ SYT ( η ) . W e write M T for the noncrossing matching associated to T , and W T for the degree two sl k web associated to T . Through the above construction, we have shown the following. R ecall from Section 2.3 that for a standard Y oung tableau T , we denote by col i ( T ) the entries in column i of T . Also, when T has two columns, then τ ∗ ( T ) is the subset of entries j in col 1 ( T ) with j + 1 ∈ col 2 ( T ) . W e also write col 2 ( T ) = { b 1 < b 2 < · · · < b k } and [ n ] = { 1, 2, . . . , n } . Lemma 2.9. Let T ∈ SYT (( k , k ) ∗ ) with corresponding noncrossing matching M T and degree two sl k web W T . The following are equal: (i) | τ ∗ ( T ) | , if there is some i ∈ [ k − 1 ] with b i = 2 i , and | τ ∗ ( T ) | + 1 otherwise, (ii) The number of arcs of the form { i , i + 1 } mod 2 k in M T , (iii) The number of claws in W T . 11 Proof . The equality between (ii) and (iii) is immediate from the above construction, so we show equality between (i) and (ii) . Given T ∈ SYT (( k , k ) ∗ ) , we first notice that there is some i ∈ [ k − 1 ] with b i = 2 i if and only if the first i rows of T form a standard Y oung tableau of shape ( i , i ) ∗ . Indeed, there are 2 i entries in the first i rows of T and, if b i = 2 i , then because the rows and columns of T are strictly increasing, the entries in the first i rows of T are exactly { 1, 2, . . . , 2 i } . Let T ′ be the tableau consisting of these first i rows of T . Then, the corresponding noncrossing matching M T ′ is a subgraph of M T , since the construction of M T matches vertices in the second column in increasing order . In particular , in both M T ′ and M T , the vertex 1 is matched with some 2, 3, . . . , 2 i , so { 1, 2 k } = { 2 k , 2 k + 1 } mod 2 k is not an edge in M T . So we conclude that in this case, there are exactly | τ ∗ ( T ) | edges in M T of the form { i , i + 1 } mod 2 k . If there is no i ∈ [ k − 1 ] with b i = 2 i , then each b i with i < k is matched in M T with a vertex j > 1. Hence we obtain in M T the edge { 2 k , 1 } = { 2 k , 2 k + 1 } mod 2 k , and, consequently , there are exactly | τ ∗ ( T ) | + 1 edges in M T of the form { i , i + 1 } mod 2 k . ■ 3. G E O M E T RY O F S M O O T H C O M P O N E N T S In this section, we prove our main results: characterizing and describing the geometry of smooth components of two column rectangle Springer fibers in terms of the webs described in Section 2.5 . W e give the characterization in Section 3.1 and the geometric description in Section 3.2 . Then, in Section 3.3 , we discuss the relation between the P oincaré polynomials of smooth components and rotation- and reflection-invariance of degree two sl k webs. In this section, we take web to mean degree two sl k web. 3.1. Characterization of smooth components. Fix a two column rectangle partition η = ( k , k ) ∗ for some k ≥ 2, and set n = 2 k . R ecall from Section 2.2 that we write S η for the Springer fiber associated to η and S T for the component of S η corresponding to T ∈ SYT ( η ) . Also, we write W T for the degree two sl k web corresponding to T . In this subsection, we reinterpret Theorem 2.1 in terms of degree two sl k webs; see Theorem 3.2 . A graph is a forest if it has no cycles, and is a tree if it is a forest that is also connected: there is a path between any two vertices. W e say that a web is a forest when its underlying graph is a forest, and similarly for trees. In Figure 5 , the web on the left is a tree, and the web on the right is not a forest. The web in Figure 6 is a forest but not a tree. 1 2 3 4 5 6 7 8 9 1 0 F I G U R E 6. A degree two sl 5 web that is a forest but not a tree. Lemma 3.1. Let T ∈ SYT (( k , k ) ∗ ) . Then, W T is a forest if and only if it has at most 3 claws. 12 Proof . W e leverage the construction in Section 2.5 . That is, consider the weighted triangulation of the polygon P constructed from the noncrossing matching M T . Because k ≥ 2, the polygon P has at least two vertices. The number of filled internal vertices in W T is exactly the number of faces in a triangulation of P . It follows from Lemma 2.9 that the number of vertices of P is exactly the number of edges of the form { i , i + 1 } mod 2 k in the noncrossing matching M T . So we claim that W T is a forest if and only if P is a line or a triangle. The converse is clear , so suppose that the polygon P has at least four vertices and consider any triangulation of P . T ake any two vertices u , v of P that are adjacent by an internal edge e in the triangulation, that is, an edge that is not on the boundary of the polygon in the weighted dissection. Then, both u and v are incident to the two distinct faces incident to e , say f , g . Consequently , the vertices in W T corresponding to u and v are both adjacent to the filled internal vertices corresponding to f and g , which gives a 4-cycle in W T . ■ W e have the following graph-theoretic reinterpretation of [ FM11 , Theorem 1.2 ] . Theorem 3.2. Let η = ( k , k ) ∗ and T ∈ SYT ( η ) . Consider the component S T of S η associated to T and the degree two sl k web W T associated to T . Then, S T is smooth if and only if W T is a forest. Proof . In our setting, Theorem 2.1 says the component of S η associated to T is smooth if and only if | τ ∗ ( T ) | ≤ 3 and, when equality holds, we moreover have that there is some i ∈ [ k − 1 ] such that b i = 2 i . This occurs, by Lemma 2.9 , if and only if M T has at most three arcs of the form { i , i + 1 } mod 2 k . Lemma 3.1 guarantees that this is equivalent to W T being a forest. ■ Example 3.3. Consider the webs in Figure 1 . The web on the left is a forest, hence corresponds to a smooth component of the associated Springer fiber . In contrast, the web on the right is not a forest, so its corresponding component is singular . The webs in Figure 5 are in the same move equivalence class (as in [ GPPSS25 ] ), so correspond to the same component of the associated Springer fiber , and this component is singular . All of the webs in Figures 6 and 7 correspond to smooth components. As a consequence of Theorem 3.2 , we give the following enumeration of the number of smooth components of the two column rectangle Springer fiber S ( k , k ) ∗ . This result brought to our attention an error in an enumeration of Mansour [ Man25 ] , which we have corrected [ CM26 ] . In Figure 7 , we illustrate the 5 degree two sl 3 webs, all of which correspond to smooth components of S ( 2,2,2 ) . 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 F I G U R E 7. The 5 degree two sl 3 webs. Corollary 3.4. The number of smooth components of S ( k , k ) ∗ is k + 2  k 3  , for any k ≥ 2 . W e give two proofs of Corollary 3.4 . 13 First proof of Corollary 3.4 . Let W be a degree two sl k web with boundary vertices 1, 2, . . . , 2 k . W e call the arc of the boundary of the disc between vertices j and j + 1 a break if j and j + 1 are in different claws of W . Notice that W is determined uniquely by its breaks. Indeed, since W has no cycles, it is the unique element in its move equivalence class. If W has exactly two claws, then the breaks occur at ( j , j + 1 ) and ( j + k , j + k + 1 ) , because the internal vertices of W have degree k . Hence there are k webs with exactly two claws. Consider now webs with exactly three claws. Each unfilled internal vertex incident to a claw has degree k and is adjacent to the filled internal vertex, hence is adjacent to at most k − 1 boundary vertices. Consequently , the distance between consecutive breaks is at most k − 1. W e construct two webs with exactly three claws by picking three breaks among the arcs of the boundary of the disc ( 1, 2 ) , ( 2, 3 ) , . . . , ( k , k + 1 ) . There are  k 3  ways to make this selection and each selection gives rise to exactly two webs with three claws: one by moving the middle break antipodally , and one by moving the first and third breaks antipodally . The desired formula follows. Our observation that the distance between consecutive breaks is at most k − 1 guarantees that we obtain every three-claw web in this way . ■ Second proof of Corollary 3.4 . There are k degree two sl k webs with exactly two claws. Indeed, for each i ∈ [ k ] , the claws in the corresponding web have vertices { i , i + 1, . . . , i + k − 1 } and { i + k , . . . , 2 k , 1, . . . , i − 1 } . So it remains to show that the number of webs with exactly three claws is exactly 2  k 3  . W e identify a three-claw web with its three breaks between claws, as defined in the first proof of Corollary 3.4 . R ecall that there is a maximum distance of k between consecutive breaks. F or the first break, there are 2 k choices. The second break can be placed at anywhere with distance in { 2, 3, . . . , k − 1 } from the first break. There are two choices for the second break at each of these distances. If the second break is at distance i from the first, then there are i − 1 choices for the remaining break for the third break to be within distance k − 1 of both the first and second breaks. So, the number of three-claw webs is 1 3!  2 k k − 1 X i = 2 2 ( i − 1 )  = 2 3!  2 k k − 2 X i = 1 i  = 2 3!  2 k · ( k − 1 )( k − 2 ) 2  = 2  k 3  . ■ In Schubert calculus, singularities and other properties are often characterized in terms of pattern avoidance conditions. Consider permutations v ∈ S m and w ∈ S k with m ≤ k . W rite v and w in one-line notation v = v 1 v 2 · · · v m and w = w 1 w 2 · · · w k , so that v ( j ) = v j and w ( i ) = w i . W e say that w contains the pattern v if there is a substring w i 1 · · · w i m of w such that w i j < w j ℓ if and only if v j < v ℓ for all j , ℓ . When this occurs, the entries in such a substring w i 1 · · · w i m have the same relative order as v . If w does not contain v , then we say that w avoids the pattern v , or that w is v -avoiding. F or instance, the permutation 352614 contains the pattern 2143, while the permutation 356214 avoids 2143. W e have the following. Corollary 3.5. Let η = ( k , k ) ∗ with k ≥ 3 . The smooth components of S η are equinumerous with the permutations in S k that avoid all of the patterns 321 , 2143 , and 3124 . Proof . Since k + 2  k 3  = k + k ( k − 1 )( k − 2 ) 3 = k 3 − 3 k 2 + 5 k 3 , the result follows from combining Corollary 3.4 and [ OEIS , A116731 ] . ■ 14 It would be interesting to explicitly describe the correspondence between these smooth components and pattern-avoiding permutations. P ermutations that avoid 321 are called fully commutative and appear throughout algebraic combinatorics; see e.g., [ BJS93 ; Ste96 ; Gre09 ; BHY19 ] . The permutations avoiding 2143 are called vexillary and also appears in both alge- braic combinatorics and Schubert calculus [ LS82 ; KMY09 ; And19 ] . F or instance, the Schubert variety X w is smooth if and only if w avoids both 2143 and 1324. W e refer the reader to [ AB16 ] for more on properties characterized by pattern avoidance. R emark 3.6. The number of components of S ( k , k ) ∗ is the number of standard Y oung tableaux of shape ( k , k ) ∗ . This value is 1 k + 1  2 k k  , the k -th Catalan number (see, e.g., [ Sta15 , p. 44 ] ). From this and Corollary 3.4 , the fraction of components of S ( k , k ) ∗ that are smooth tends exponentially towards 0 as k → ∞ . 3.2. Geometry from degree two sl k webs. In the previous subsection, we characterized the smooth components of the Springer fiber S ( k , k ) ∗ in terms of degree two sl k webs. The purpose of this subsection is to use these webs to describe the geometry of the smooth components. These components are iterated fiber bundles , as in Definition 2.2 . Let T ∈ SYT (( k , k ) ∗ ) and write W T for the corresponding degree two sl k web. W e define the first claw of W T to be the first claw clockwise from the arc ( 2 k , 1 ) of the boundary disc that does not contain the vertex 2 k . That is, if 1 and 2 k are in different claws, then the first claw is the claw containing 1; otherwise, it is the first claw strictly clockwise of the claw containing 1. The second claw is the next claw appearing clockwise of the first claw , and so on. Example 3.7. First consider the webs in Figure 1 . The web on the left has first claw { 2, 3, . . . , 8 } , while the web on the right has first claw { 1, 2, . . . , 5 } . R ecall that both webs in Figure 5 are in the same move equivalence class (as discussed in Section 2.5 ), and they have the same first claw , namely , the vertices { 3, 4, 5 } . The following is a graph-theoretic reinterpretation of [ FMSO15 , Theorem 3 ] . Theorem 3.8. Suppose that the degree two sl k web W is a forest, hence corresponds to a smooth component S W of the Springer fiber S ( k , k ) ∗ . Then, S W is an iterated fiber bundle whose base is given by the following cases. • If W is disconnected, then let i be the maximum integer such that the vertices 1, 2, . . . , i are in the same claw of W . The base is  Fl ( i ) × Fl ( k ) , P i , P i + 1 , . . . , P k − 1  . • Otherwise W is connected. Let i , j , and m = 2 k − i − j be the number of vertices in the first, second, and third claws of W , respectively . Let ℓ be the multiplicity of the edge between the filled internal vertex and the unfilled vertex of second claw . Then, the base is  Fl ( i ) × Fl ( j ) , Gr ℓ ( m ) , P k − m , P k − m + 1 , . . . , P k − 1  . Proof . Let T ∈ SYT (( k , k ) ∗ ) be the standard Y oung tableau for which W = W T under the con- struction in Section 2.5 . R ecall that M T denotes the noncrossing matching corresponding to T . Suppose first that W is disconnected, or equivalently , has exactly two claws. By Lemma 2.9 , we have that M T has exactly two arcs of the form { i , i + 1 } mod 2 k . W e have the following subcases. If { 1, 2 k } is an edge in M T , then τ ∗ ( T ) = { α } by Lemma 2.9 . Since the internal vertices of W have the same degree, it follows that α = k . So W has claws with vertices { 1, . . . , k } and 15 { k + 1, . . . , 2 k } . From Definition 2.4 , we set a T = 0, b T = k , and c T = 0. Then, Theorem 2.5 guarantees that the component of S ( k , k ) ∗ corresponding to T is an iterated fiber bundle with base Fl ( k ) 2 , as desired. Otherwise, τ ∗ ( T ) = { α < β } . The claws in W T correspond to vertices { α + 1, . . . , β } and { β + 1, . . . , 2 k , 1, . . . , α } . Again, the internal vertices have degree k , so these vertex sets have equal cardinality , both containing exactly k vertices. In particular , k = β − α . Now define a T , b T , c T as in Definition 2.4 and notice that i = α . Then, a T = k − ( β − α ) = 0, a T + b T = α = i , b T + c T = β − α = k , and b T = β − k = i . So from Theorem 2.5 , we obtain that the component of S ( k , k ) ∗ corresponding to T is an iterated bundle with base  Fl ( i ) × Fl ( k ) , P i , P i + 1 , . . . , P k − 1  , as desired. W e now treat the second case, when W is connected, so has exactly three claws. Again, we have subcases depending on whether or not { 1, 2 k } is an edge in the noncrossing matching M T . Suppose first that { 1, 2 k } is an edge in M T , so by Lemma 2.9 we have that τ ∗ ( T ) = { α < β } . Consequently , the claws in W T contain the boundary vertices { 1, . . . , α } , { α + 1, . . . , β } , and { β + 1, . . . , γ } . Hence, with i and j as in the statement of the theorem, we have that i = α and j = β − α . From this it follows that m = 2 k − β and k − m = β − k . Note that from the construction of the interior of the web in Section 2.5 , the value ℓ is exactly the number of edges in M T with one endpoint in each of the vertex sets { 1, . . . , α } and { β + 1, . . . , 2 k } . The noncrossing matching M T is a perfect matching on 2 k vertices, so has exactly k edges. So ℓ is the difference between k and the number of edges incident to one of the vertices in { α + 1, . . . , β } . That is, ℓ = k − ( β − α ) . W e have computed that a T = ℓ , a T + b T = α = i , b T + c T = β − α = j , and, a T + c T = 2 k − β = m , b T = β − k = k − m . The desired base for the bundle follows from Theorem 2.5 . Lastly , suppose that { 1, 2 k } is not an edge in M T , so τ ∗ ( T ) = { α < β < γ } where γ < 2 k . The claws in W T now have corresponding vertices { α + 1, . . . , β } , { β + 1, . . . , γ } , and { γ + 1, . . . , 2 k , 1, . . . , α } . The sizes i and j of the first two claws are i = β − α and j = γ − β , hence i + j = γ − α . Similar to the previous case, we have that ℓ = k − ( γ − β ) . So, we have that a T = k − ( γ − β ) = ℓ , a T + b T = β − α = i , b T + c T = γ − β = j , and, a T + c T = 2 k − ( γ − α ) = 2 k − i − j , b T = ( γ − α ) − k = 2 k − m − k = k − m . The desired result now follows from Theorem 2.5 . ■ Example 3.9. Consider the web W in Figure 6 . This web W has exactly two connected com- ponents, so Theorem 3.8 says that the component of the Springer fiber S ( 5,5 ) ∗ corresponding to W is an iterated fiber bundle with base  Fl ( 3 ) × Fl ( 5 ) , P 3 , P 4  . The component of S ( 8,8 ) ∗ corresponding to the connected web in the left of Figure 1 is an iterated fiber bundle with base  Fl ( 7 ) × Fl ( 3 ) , Gr 5 ( 6 ) , P 2 , P 3 , . . . , P 7  . 16 Every component of a two row Springer fiber is smooth [ Fun03 ] . In this setting, both the geometry and topology are understood, and moreover , we have descriptions of the pairwise intersections of components; see, e.g., [ Fun03 ; R us11 ; SW12 ] . It would be interesting to obtain, in the two column case, descriptions of the singular components and results analogous to the two row case. 3.3. P oincaré polynomials for smooth components. A degree two sl k web inherits a natural action of the dihedral group by rotation and reflection of its underlying graph (leaving the ver- tex labels fixed). This corresponds to the procedures of promotion and evacuation , respectively , on the corresponding standard Y oung tableau; see [ PPR09 ; Fra23 ; PP23 ] . In this section, we will show that for smooth components of the Springer fiber S ( k , k ) ∗ , the P oincaré polynomial, de- fined in Section 2.3 , is invariant under this dihedral action. Moreover , the P oincaré polynomial of a smooth component uniquely identifies the corresponding dihedral orbit of the webs. 1 2 3 4 5 6 1 2 3 4 5 6 F I G U R E 8. A web W (left) and its clockwise rotation ρ · W (right). Let W be a degree two sl k web that is a forest, and hence corresponds to a smooth component of S ( k , k ) ∗ , and recall that we denote by S W this component. Denote by P W ( q ) , or simply P W when there is no ambiguity , the P oincaré polynomial of S W . Also, let D 2 k = 〈 r , s | r 2 k = s 2 = 1, r s = s r − 1 〉 be the dihedral group of a regular 2 k -gon, so | D 2 k | = 4 k . W e fix ρ ∈ D 2 k to be the rotation such that ρ · W is a clockwise rotation of the underlying graph of W ; see Figure 8 . W e have the following graph-theoretic reinterpretation of [ FMSO15 , Theorem 4 ] . Theorem 3.10. Let W and W ′ be degree two sl k webs that are both forests. Then, the P oincaré polynomials of S W and S W ′ are equal if and only if W = σ · W ′ for some σ ∈ D 2 k . R emark 3.11. R eflection of a web corresponds to evacuation on the associated tableau. F or a tableau T of any shape, the components of the Springer fiber corresponding to T and its evacuation are isomorphic [ vLee00 , Corollary 3.4 ] (see also [ KP25 , Theorem 7.4 ] ). W e obtain by other methods—namely , a direct computation using Theorem 3.8 —the equality of P oincaré polynomials for components corresponding to evacuation-equivalent tableaux, in the two col- umn rectangle case. Although there is an isomorphism between components of a two column rectangle Springer fiber whose webs are related by reflection, the same does not hold for rotation. F or instance, consider the Springer fiber S ( 2,2 ) . There are two standard Y oung tableaux of shape ( 2, 2 ) ; see Figure 9 , which also shows that their corresponding webs are in the same rotation orbit. Y et [ SW12 , Example 8 ] shows that the corresponding components are not isomorphic: The first is a copy of P 1 × P 1 , while the second is a nontrivial P 1 -bundle over P 1 (in particular , it is a Hirzebruch surface). 17 1 2 3 4 1 2 3 4 1 3 2 4 1 2 3 4 F I G U R E 9. The standard Y oung tableaux of shape ( 2, 2 ) and their corresponding degree two sl 2 webs. However , these components of S ( 2,2 ) are homeomorphic as topological spaces: All compo- nents of a two row Springer fiber are homeomorphic, as shown in [ CK08 , Theorem 2.1 ] , build- ing on [ Kho04a ] . (F or more details on this argument, see [ RT11 , Appendix ] .) It is unclear to the author whether the components in the two column rectangle case with equal P oincaré polynomials are homeomorphic. The proof of the forward direction of Theorem 3.10 is a computation that leverages the description of the geometry in Theorem 3.8 . W e use in the following proof the P oincaré poly- nomials given in Fact 2.6 of projective space, the Grassmannian, and the flag variety . Proof of the forward direction of Theorem 3.10 . Suppose that W = σ · W ′ for some σ ∈ D 2 k . W e first treat the case where the webs W and W ′ are both disconnected. W e make two obser- vations. First, in this case, reflection invariance follows from rotation invariance. That is, if σ ∈ D 2 k is a reflection, then there is some rotation ˜ ρ ∈ D 2 k such that σ · W = ˜ ρ · W . Secondly , for rotation invariance, it suffices to show invariance in the case of W = ρ · W ′ with ρ defined as in the discussion preceding Theorem 3.10 . Indeed, any rotation in D 2 k is of the form ρ t for some t . So suppose that W = ρ · W ′ . Denote by i W the integer i in Theorem 3.8 for W , and similarly , i W ′ for that of W ′ . If i W = i W ′ + 1, then using Theorem 3.8 , P W = [ i W ] ! [ k ] ! [ i W + 1 ][ i W + 2 ] · · · [ k ] = [ i W + 1 ] ! [ k ] ! [ i W + 2 ][ i W + 3 ] · · · [ k ] = P W ′ , as desired. Otherwise suppose that i W  = i W ′ + 1, which implies that i W = 1 and i W ′ = k . Then, P W =  [ 1 ][ k ] !  [ 2 ][ 3 ] · · · [ k ] =  [ k ] !  2 = P W ′ . Hence the desired result holds when W and W ′ are both disconnected. W e next show that the P oincaré polynomial is rotation invariant for webs W and W ′ that are both connected. Again, it is sufficient to consider the case where W = ρ · W ′ . W e proceed by cases. Suppose that the boundary vertex 1 is in the same claw of W and W ′ , and is not the first vertex in the first claw of W . Then the values i , j , and ℓ as in Theorem 3.8 are exactly the same for both W and W ′ , and the desired result is immediate. This argument also applies in the case that 1 appears in different claws of W and W ′ . So suppose that 1 appears in the same claw of W and W ′ and is the first vertex of the first claw of W . Let i W , j W , ℓ W , m W be as in Theorem 3.8 for W , and similarly i W ′ , j W ′ , ℓ W ′ , m W ′ for W ′ . It then follows that i W = m W ′ , j W = i W ′ , and m W = j W ′ . Also, ℓ W = k − j W and ℓ W ′ = k − j W ′ . 18 W e then compute that P W ′ = [ i W ′ ] ! [ j W ′ ] !  m W ′ ℓ W ′  [ k − m W ′ + 1 ] · · · [ k − 1 ][ k ] = [ j W ] ! [ m W ] !  i W k − m W  [ k − i W + 1 ] · · · [ k − 1 ][ k ] = [ i W ] ! [ j W ] ! [ m W ] ! [ k ] ! [ k − m W ] ! [ k − j W ] ! [ k − i W ] ! , where for the last equality , we use the fact that i W − ( k − m W ) = k − j W after substituting m W = 2 k − i W − j W . Continuing, and using the observations that ℓ W = k − j W and k − i W = ( 2 k − i W − j W ) − ( k − j W ) = m W − ℓ W , the desired result now follows: P W ′ = [ i W ] ! [ j W ] ! [ m W ] ! [ k ] ! [ k − m W ] ! [ k − j W ] ! [ k − i W ] ! = [ i W ] ! [ j W ] !  m W ℓ W  [ k − m W + 1 ] · · · [ k − 1 ][ k ] = P W . Lastly we treat reflection invariance in the connected case. Consider any reflection σ and take W and W ′ such that W = σ · W ′ . Using the rotation invariance, we freely assume that the second claws of W and W ′ are the same. Hence, we have that i W = m W ′ , j W = j W ′ , m W = i W ′ , and ℓ W = ℓ W ′ . Then, P W = [ i W ] ! [ j W ] !  m W ℓ W  [ k − m W + 1 ] · · · [ k − 1 ][ k ] = [ i W ] ! [ j W ] ! [ m W ] ! [ k ] ! [ ℓ W ] ! [ m W − ℓ W ] ! [ k − m W ] ! . Since W is k -valent, we have that j W + ℓ W = k . From this, we observe that k − m W = 2 k − m W − ( j W + ℓ W ) = ( 2 k − i W ′ − j W ′ ) − ℓ W ′ = m W ′ − ℓ W ′ , and analagously , m W − ℓ W = k − m W ′ . With this, we compute that P W = [ i W ′ ] ! [ j W ′ ] ! [ m W ′ ] ! [ k ] ! [ ℓ W ′ ] ! [ m W ′ − ℓ W ′ ] ! [ k − m W ′ ] ! = [ i W ′ ] ! [ j W ′ ] !  m W ′ ℓ W ′  [ k − m W ′ + 1 ] · · · [ k − 1 ][ k ] = P W ′ , which completes the proof of the forward direction. ■ F or the reverse direction, we rely on Theorem 2.7 . T o that end, we require the following lemma that describes the relation between the number of claws of a degree two sl k web W and the values a T , b T , and c T in Definition 2.4 when T is the tableau associated to W . Lemma 3.12. Let W be a degree two sl k web corresponding to a tableau T , and assume that W is a forest. Define a T , b T , and c T as in Definition 2.4 . Then, W is connected if and only if both a T and c T are nonzero. Proof . Denote by S T the component of the Springer fiber corresponding to T . Then, a T = 0 or c T = 0 if and only if , by Theorem 2.5 , S T is an iterated fiber bundle with base  Fl ( a T + b T ) × Fl ( b T + c T ) , P b T , P b T + 1 , . . . , P k − 1  . (In particular , the Grassmannian is a point and is omitted; see R emark 2.3 .) This is equivalent, by Theorem 3.8 , to W being disconnected. Otherwise, W is connected. ■ 19 Proof of the reverse direction of Theorem 3.10 . Suppose that we have equality of P oincaré poly- nomials P W = P W ′ . Let T and T ′ be the standard Y oung tableaux corresponding to W and W ′ , respectively . Define a T , b T , c T and a T ′ , b T ′ , c T ′ as in Definition 2.4 . By Theorem 2.7 we have the following two cases. Suppose first that 0 ∈ { a T , c T } and that 0 ∈ { a T ′ , c T ′ } . Lemma 3.12 says that both W and W ′ have exactly two claws, so are in the same D 2 k orbit. Otherwise, we have { a T , b T , c T } = { a T ′ , b T ′ , c T ′ } as sets, and that a T , c T , a T ′ , and c T ′ are all nonzero. From the rotation- and reflection-invariance of the P oincaré polynomial, it suffices to show that the set of claw sizes of W T and W T ′ are equal. By Lemma 3.12 , both W T and W T ′ have three claws. R ecall from Lemma 2.9 that the web W T has three claws if and only if • | τ ∗ ( T ) | = 2 and there is no i ∈ [ k − 1 ] such that b i = 2 i , or , • | τ ∗ ( T ) | = 3 and there is some i ∈ [ k − 1 ] such that b i = 2 i . Suppose that T is in the first case, so τ ∗ ( T ) = { α < β } . Then—using Lemma 2.9 again— the first claw of T involves boundary vertices { 1, 2, . . . , α } , and k − c T = α . The second claw involves vertices { α + 1, . . . , β } , and hence has size β − α . Notice that k − a T = β − α . In particular , the sizes of the claws are determined by a T , b T , c T . Now take T to be in the second case, so τ ∗ ( T ) = { α < β < γ } . Then from Lemma 2.9 , the first claw is not incident to the boundary vertex 1, and instead involves vertices { α + 1, . . . , β } . This claw size is given by k − c T = β − α . Similarly , the second claw contains vertices { β + 1, . . . , γ } , and k − a T = γ − β gives the corresponding size. Similarly , the sizes of the claws of W ′ are completely determined by a T ′ , b T ′ , and c T ′ . So since { a T , b T , c T } = { a T ′ , b T ′ , c T ′ } , we conclude that W and W ′ have claws of the same sizes, and hence lie in the same dihedral orbit. This completes the proof. ■ Example 3.13. Consider the webs in Figure 7 . Let P W i be the P oincaré polynomial of the component of S ( 2,2,2 ) corresponding to the i -th web W i (counted from left-to-right). It is a straightforward computation using Fact 2.6 and Theorem 3.8 to see that P W 1 = P W 2 = P W 3 = ([ 3 ] ! ) 2 and P W 4 = P W 5 = [ 2 ] 4 [ 3 ] . A P P E N D I X A. G E O M E T RY F R O M N O N C R O S S I N G M A T C H I N G A N D R A Y D I A G R A M S In Section 3 , we used webs to characterize and describe the geometry of the smooth compo- nents of two column rectangle Springer fibers. The results of Fresse, Melnikov , and Sakas-Obeid [ FM11 ; FMSO15 ] extend to the arbitrary , not necessarily rectangular , two column case. How- ever , outside of some special cases [ PPS22 ; Kim24 ; FPPS25 ] , there are no webs corresponding to non-rectangular two column tableaux, so our results do not naturally generalize within the framework of degree two sl k webs. On the other hand, there is a combinatorially natural bijection between two row tableaux and noncrossing matching and ray diagrams [ R us11 ; SW12 ] . These diagrams are used to de- scribe the geometry and topology of two row Springer fibers. In this appendix, we show that these diagrams can also be used to characterize and describe the geometry of smooth compo- nents of two column Springer fibers, even though these diagrams are not two column webs. This construction relies on tableaux conjugation, which is algebraically subtle. W e believe that this subtlety arises in the characterization and description of the geometry of the smooth components is not as straightforward as in the two column rectangle and two row cases. 20 A.1. Noncrossing matching and ray diagrams. A noncrossing matching and ray diagram with vertices V = { 1, 2, . . . , n } and k matching edges is a planar diagram such that the vertices are drawn in order along a horizontal baseline, 2 k vertices are matched by noncrossing edges drawn above the baseline, and the remaining n − 2 k vertices each have an incident ray . W e draw a ray as an edge with one endpoint that extends off with infinite height, so that for all matching edges { i , ℓ } and any vertex j incident to a ray , we have either j < i or k < j . 1 3 2 5 4 1 2 3 4 5 1 2 3 5 4 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 4 2 5 3 1 2 3 4 5 1 3 2 4 5 1 2 3 4 5 F I G U R E 10. The standard Y oung tableaux of shape ( 2, 2, 1 ) and their correspond- ing noncrossing matching and ray diagrams. The bijection between two column standard Y oung tableaux and noncrossing matching and ray diagrams is a slight modification to the bijection between two column rectangle standard Y oung tableaux and noncrossing (perfect) matchings. The bijection is given as follows. Fix a tableau T ∈ SYT (( n − k , k ) ∗ ) , where 1 ≤ k ≤ n / 2. W e construct a noncrossing matching and ray diagram with k matching arcs and n − 2 k rays. Denote by { b 1 < b 2 < · · · < b k } the entries in the second column of T . Match b 1 with the largest entry in the first column that is less than b 1 , then match b 2 with the largest remaining unmatched entry in the first column less than b 2 , and so on. After completing the matching, from each remaining unmatched vertex draw a ray . Figure 10 shows the five standard Y oung tableaux of shape ( 3, 2 ) ∗ and their corresponding noncrossing matching and ray diagrams. A.2. Characterization of smooth components. In this section, we give a characterization of the smooth components of two column Springer fibers in terms of noncrossing matching and ray diagrams, analogous to Section 3.1 . T o begin, we record the arbitrary two column characterization of smooth components of Fresse and Melnikov in terms of standard Y oung tableaux. Our notation is as in Section 2.3 . Recall that for a two column tableau T , we denote by τ ∗ ( T ) the entries j in the first column of T for which j + 1 is in the second column of T . Also recall that we denote by b 1 < b 2 < · · · < b k the entries in the second column of T . The following is the full version of Theorem 2.1 . Theorem A.1 ( [ FM11 , Theorem 1.2 ] ) . Let T ∈ SYT (( n − k , k ) ∗ ) , where 1 ≤ k ≤ n / 2 . Then, the component S T of S ( n − k , k ) ∗ corresponding to T is smooth if and only if one of the following holds: (S1) | τ ∗ ( T ) | = 1 , (S2) | τ ∗ ( T ) | = 2 and either b k = n or b i = 2 i for some i ∈ [ k ] , or , (S3) | τ ∗ ( T ) | = 3 , b k = n, and b i = 2 i for some i ∈ [ k − 1 ] . Let M be a noncrossing matching and ray diagram with n vertices and n − 2 k rays. Denote by S M the component of the Springer fiber corresponding to M under the bijection in Section A.1 and Theorem 1.1 . W e say that a short edge is an edge of the form { i , i + 1 } for some i < n . 21 The following is a diagrammatic reinterpretation of [ FM11 , Theorem 1.2 ] . Theorem A.2. Let M be a noncrossing matching and ray diagram with n vertices and n − 2 k rays. Then, the component S M of the Springer fiber S ( n − k , k ) ∗ corresponding to M is smooth if and only if (S1 ′ ) M has exactly 1 short edge; (S2 ′ ) M has exactly 2 short edges, but at least one of the vertices 1 , n is not a ray in M ; or , (S3 ′ ) M has exactly 3 short edges, but { 1, n } is not an edge of M , and neither 1 nor n is a ray . Proof . W e show the slightly stronger result that the conditions (S1 ′ ) , (S2 ′ ) , and (S3 ′ ) are equiv- alent to, respectively , the conditions (S1) , (S2) , and (S3) in Theorem A.1 . Let T be the tableau corresponding to M under the bijection in Section A.1 . In contrast with Lemma 2.9 , we have that | τ ∗ ( T ) | is exactly equal to the number of short edges in M . So conditions (S1) and (S1 ′ ) are equivalent. First, note that n appears in the last box in either the first or second column. In particular , n appears in the second column if and only if n is not a ray . Let b 1 < b 2 < · · · < b k be the entries in the second column of T . W e observed in the proof of Lemma 2.9 that b i = 2 i for some i ∈ [ k ] if and only if the first i rows of T form a standard Y oung tableau of shape ( i , i ) ∗ . Since 1 is matched in M if and only if there is such an i ∈ [ k ] , the conditions (S2) and (S2 ′ ) are equivalent. F or the third pair of conditions, note that (S3 ′ ) prohibits i = k , which is equivalent to forbidding { 1, n } to be an edge in the matching. This shows that (S3) and (S3 ′ ) are equivalent, which completes the proof . ■ W e remark that since matching edges cannot pass over rays, the condition in (S3 ′ ) that { 1, n } is not an edge of M is vacuous unless the corresponding partition is a two column rectangle. 1 3 2 4 5 7 6 8 1 2 3 4 5 6 7 8 1 3 2 6 4 7 5 8 1 2 3 4 5 6 7 8 1 2 3 5 4 8 6 7 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 F I G U R E 11. F our standard Y oung tableaux of shape ( 5, 3 ) ∗ and their correspond- ing noncrossing matching and ray diagrams. Example A.3. All of the noncrossing matching and ray diagrams in Figure 10 correspond to smooth components of the Springer fiber S ( 3,2 ) ∗ . In Figure 11 , the components corresponding to the noncrossing matching and ray diagrams on the left are smooth, while those on the right are singular . W e leave it to the reader to verify with Theorem A.1 the same conclusions hold from the corresponding tableaux. 22 A.3. Geometry from noncrossing matching and ray diagrams. The goal of this section is to use noncrossing matching and ray diagrams to give a result analogous to Theorem 3.8 , which describes the geometry of the smooth components of two column rectangle Springer fibers. In analogy with the claws of degree two sl k webs, whose sizes describe the geometry of smooth components of two column rectangle Springer fibers, we make the following definition. Definition A.4. Let M be a noncrossing matching and ray diagram on n vertices. Suppose that i < j are such that { i , i + 1 } and { j , j + 1 } are short edges in M . The ( i , j )-pseudoclaw of M is the set of matching edges and rays of M that do not have an endpoint in { i + 1, i + 2, . . . , j } . F or the analogy with the degree two sl k webs, notice that we have a claw in Figure 5 corre- sponding to the vertices 10, 1, 2, which are exactly the vertices in the (2, 9)-pseudoclaw of the matching in Figure 3 . Example A.5. Consider the noncrossing matching and ray diagrams in Figure 11 . The (3, 6)- pseudoclaw in the top-left diagram consists only of the ray from vertex 8. F or the lower-left diagram, the (5, 7)-pseudoclaw consists of the matching edge { 1, 2 } and the ray from vertex 3. On the other hand, in the same diagram, the (2, 7)-pseudoclaw is empty . W e give the following diagrammatic reinterpretation of [ FMSO15 , Theorem 3 ] . Theorem A.6. Let M be a noncrossing matching and ray diagram with n vertices, k matching edges, and r = n − 2 k rays. Suppose that M corresponds to a smooth component S M of S ( n − k , k ) . Then S M is an iterated fiber bundle with base given by the following cases: • If M has exactly one short edge, say { i , i + 1 } , then let ℓ be the number of rays incident to vertices at most i . Then, the base is given by  Fl ( i ) × Fl ( n − i ) , Gr ℓ ( r )  . • Suppose that M has exactly two short edges, denoted { i , i + 1 } and { j , j + 1 } with i < j . Let ℓ be the size of the ( i , j )-pseudoclaw of M . If n is a ray , then let m = r + i , and the base is  Fl ( n − j ) × Fl ( j − i ) , Gr ℓ ( m ) , P k − i , P k − i + 1 , . . . , P k − 1  . Otherwise n is not a ray , then let m = r + ( n − j ) , where n − j is the number of vertices strictly larger than j . The base is  Fl ( i ) × Fl ( j − i ) , Gr ℓ ( m ) , P j − ( r + k ) , P j − ( r + k )+ 1 , . . . , P k − 1  . • If M has exactly three short edges, say { i , i + 1 } , { j , j + 1 } , and { h , h + 1 } , with i < j < h, then take ℓ to be the size of the ( j , h)-pseudoclaw of M . Also let m = n + r − ( h − i ) . The base is  Fl ( j − i ) × Fl ( h − j ) , Gr ℓ ( m ) , P ( h − i ) − ( r + k ) , P ( h − i ) − ( r + k )+ 1 , . . . , P k − 1  . W e emphasize that each value appearing in the above theorem can be read off the noncross- ing matching and ray diagram. At the same time, the description is not as straightforward as Theorem 3.8 in the two column rectangle case. W e view this as a geometric motivation for the development of webs corresponding to arbitrary two column tableaux. Our proof will again leverage Theorem 2.5 . However in one case, the definitions of a T , b T , and c T differ slightly from the rectangular case given in Definition 2.4 . Definition A.7 ( [ FMSO15 , §2.4 ] ) . Let T ∈ SYT smooth (( n − k , k ) ∗ ) for some k ≤ n . Define a T , b T , and c T as in Definition 2.4 unless both | τ ∗ ( T ) | = 2 and n appears in the first column of T . In this case, let τ ∗ ( T ) = { α < β } , and set a T = ( n − k ) − ( β − α ) , b T = k − α , and c T = β − k . 23 Proof of Theorem A.6 . Let T be the standard Y oung tableaux of shape ( n − k , k ) ∗ associated to M . In the proof of Theorem A.2 , we observed that | τ ∗ ( T ) | is equal to the number of short edges in M . So, we proceed by cases on | τ ∗ ( T ) | . Suppose first that | τ ∗ ( T ) | = 1, so we have exactly one edge { i , i + 1 } between consecutive vertices. It follows that τ ∗ ( T ) = { i } . From Definition A.7 , we set a T = i − k , b T = k , and c T = n − k − i . Since k is the number of matching edges, it follows from | τ ∗ ( T ) | = 1 that each of the vertices i , i − 1, . . . , i − ( k − 1 ) , which appear in the first column of T , are matched with vertices i + 1, i + 2, . . . , i + k , respectively . In particular , a T = i − k is exactly the number of vertices less that i that are rays in M . Now , a T + b T = i , and b T + c T = n − i , while a T + c T = n − 2 k is the number of rays in M . So the desired result follows from Theorem 2.5 . Now assume that | τ ∗ ( T ) | = 2, so τ ∗ ( T ) = { i < j } . R egardless of whether n is a ray or not, we define a T = ( n − k ) − ( j − i ) = ( r + k ) − ( j − i ) since r = n − 2 k . Since { i , i + 1 } and { j , j + 1 } are the only short edges, any edge incident to a vertex v ∈ { i + 1, . . . , j − 1 } is either a ray , or a matching edge whose other endpoint w satisfies w < i or w > j + 1. Consequently , a T = ( r + k ) − ( j − i ) is exactly the size ℓ of the ( i , j )-pseudoclaw of M . If n is a ray in M , then n appears in the first column of T . So from Definition A.7 we set b T = k − i and c T = j − k . Then, a T + b T = n − j , b T + c T = j − i , and a T + c T = r + i . Theorem 2.5 now gives the claimed formula. Otherwise n is not a ray , so n appears in the second column of T . Definition A.7 says to define b T = j − ( r + k ) and c T = ( n − k ) − i , from which it follows that a T + b T = i , b T + c T = j − i , and a T + c T = r + ( n − j ) . The base of the iterated bundle given in Theorem 2.5 agrees with the desired expression. Lastly , suppose that | τ ∗ ( T ) | = 3, so τ ∗ ( T ) = { i < j < n − 1 } . F ollowing Definition A.7 , we set a T = ( n − k ) − ( h − j ) = r + k − ( h − j ) , which is exactly ℓ as before. W e also set b T = ( h − i ) − ( n − k ) = ( h − i ) − ( r + k ) and c T = ( n − k ) − ( j − i ) . Then, a T + b T = j − i , b T + c T = h − j , and a T + c T = 2 ( n − k ) − ( h − i ) = n + r − ( h − i ) . This, together with Theorem 2.5 , completes the proof . ■ Example A.8. W e saw in Example A.3 that the two webs on the left of Figure 11 correspond to smooth components of the Springer fiber S ( 5,3 ) ∗ . The diagram in the top-left of Figure 11 corresponds to a component whose base is  Fl ( 2 ) × Fl ( 4 ) , Gr 1 ( 4 ) , P , P 2  , and for the diagram in the lower-left, we have  Fl ( 3 ) 3 , Gr 2 ( 4 ) , P 1 , P 2  . 24 R E F E R E N C E S [ AB16 ] Abe, H. and S. Billey. “Consequences of the Lakshmibai-Sandhya theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry”. In: Schubert Calculus— Osaka 2012 . V ol. 71. Adv . Stud. Pure Math. Math. Soc. Japan, [ T okyo ] , 2016, 1–52. [ And19 ] Anderson, D. “ K -theoretic Chern class formulas for vexillary degeneracy loci”. In: Adv . Math. 350 (2019), 440–485. [ BHY19 ] Billey , S. C., A. E. Holroyd, and B. Y oung. “A bijective proof of Macdonald’s reduced word formula”. In: Algebr . Comb. 2.2 (2019), 217–248. [ BJS93 ] Billey , S. C., W . Jockusch, and R. P . Stanley. “Some combinatorial properties of Schubert polynomials”. In: J. Algebraic Combin. 2.4 (1993), 345–374. [ Bod22 ] Bodish, E. “W eb calculus and tilting modules in type C 2 ”. In: Quantum T opol. 13.3 (2022), 407–458. [ BW25 ] Bodish, E. and H. W u. “W ebs for the quantum orthogonal group”. In: Adv . Math. 480 (2025), P aper No. 110514, 65 pages. [ BM83 ] Borho, W . and R. MacPherson. “P artial resolutions of nilpotent varieties”. In: Analysis and topology on singular spaces, II, III (Luminy , 1981) . V ol. 101-102. Astérique. Soc. Math. France, P aris, 1983, 23–74. [ Car93 ] Carrell, J. B. “Bruhat cells in the nilpotent variety and the intersection rings of Schubert varieties”. In: J. Differential Geom. 37.3 (1993), 651–668. [ CK08 ] Cautis, S. and J. Kamnitzer. “Knot homology via derived categories of coherent sheaves. I. The sl ( 2 ) -case”. In: Duke Math. J. 142.3 (2008), 511–588. [ CHY25 ] Chen, X., W . He, and S. Y u. “Quantization of the minimal nilpotent orbits and the quantum Hikita conjecture”. In: Represent. Theory 29 (2025), 616–658. [ CM26 ] Cummings, M. and R. Mansour . Corrigendum to “Counting two-column Y oung tableaux cor- responding to smooth components of Springer fibers” [ J. Algebr . Comb. 61:18 (2025) ] . Sub- mitted. 2026. [ DKS24 ] Douglas, D. C., R. Kenyon, and H. Shi. “Dimers, webs, and local systems”. In: T rans. Amer . Math. Soc. 377.2 (2024), 921–950. [ FP16 ] F omin, S. and P . Pylyavskyy. “T ensor diagrams and cluster algebras”. In: Adv . Math. 300 (2016), 717–787. [ FKK13 ] F ontaine, B., J. Kamnitzer, and G. Kuperberg. “Buildings, spiders, and geometric Satake”. In: Compos. Math. 149.11 (2013), 1871–1912. [ Fra23 ] Fraser , C. “W ebs and canonical bases in degree two”. In: Comb. Theory 3 (2023), P aper No. 11, 26 pages. [ FLL19 ] Fraser , C., T . Lam, and I. Le. “From dimers to webs”. In: T rans. Amer . Math. Soc. 371.9 (2019), 6087–6124. [ FPPS25 ] Fraser , C., R. P atrias, O . P echenik, and J. Striker. “W eb invariants for flamingo Specht mod- ules”. In: Algebr . Comb. 8.1 (2025), 235–266. [ FP23 ] Fraser , C. and P . Pylyavskyy. “T ensor diagrams and cluster combinatorics at punctures”. In: Adv . Math. 412 (2023), P aper No. 108796, 83 pages. [ Fre09 ] Fresse, L. “Composantes singulières des fibres de Springer dans le cas deux-colonnes”. In: C. R. Math. Acad. Sci. Paris 347.11-12 (2009), 631–636. [ FM10 ] Fresse, L. and A. Melnikov. “On the singularity of the irreducible components of a Springer fiber in sl n ”. In: Selecta Math. (N.S.) 16.3 (2010), 393–418. [ FM11 ] Fresse, L. and A. Melnikov. “Some characterizations of singular components of Springer fibers in the two-column case”. In: Algebr . Represent. Theory 14.6 (2011), 1063–1086. [ FMSO15 ] Fresse, L., A. Melnikov, and S. Sakas-Obeid. “On the structure of smooth components of Springer fibers”. In: Proc. Amer . Math. Soc. 143.6 (2015), 2301–2315. 25 [ Fun97 ] Fung, F . Y . C. “On the relation between Springer fibers of the general linear group and Kazhdan-Lusztig theory”. PhD thesis. Princeton University, 1997. [ Fun03 ] Fung, F . Y . C. “On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory”. In: Adv . Math. 178.2 (2003), 244–276. [ GPPSS24 ] Gaetz, C., O . P echenik, S. Pfannerer, J. Striker, and J. P . Swanson. “Promotion permutations for tableaux”. In: Comb. Theory 4.2 (2024), P aper No. 15, 56 pages. [ GPPSS25 ] Gaetz, C., O . P echenik, S. Pfannerer, J. Striker, and J. P . Swanson. “W eb bases in degree two from hourglass plabic graphs”. In: Int. Math. Res. Not. IMRN 13 (2025), rnaf183. [ GPPSS26 ] Gaetz, C., O . P echenik, S. Pfannerer, J. Striker, and J. P . Swanson. “Rotation-invariant web bases from hourglass plabic graphs”. In: Invent. Math. 243 (2026), 703–804. [ GNST25 ] Goldwasser , T ., M. Nadeem, G. Sun, and J. T ymoczko. “Cell Closures for T wo-R ow Springer Fibers via Noncrossing Matchings”. In: Advances in the Mathematical Sciences . V ol. 38. Assoc. W omen Math. Ser . Springer , Cham, 2025, 31–81. [ Gre09 ] Green, R. M. “Leading coefficients of Kazhdan-Lusztig polynomials and fully commutative elements”. In: J. Algebraic Combin. 30.2 (2009), 165–171. [ Hig23 ] Higgins, V . “T riangular decomposition of SL 3 skein algebras”. In: Quantum T opol. 14.1 (2023), 1–63. [ HR22 ] Hopkins, S. and M. Rubey. “Promotion of Kreweras words”. In: Selecta Math. (N.S.) 28.1 (2022), P aper No. 10, 38 pages. [ IL W22 ] Im, M. S., C.-J. Lai, and A. Wilbert. “A study of irreducible components of Springer fibers using quiver varieties”. In: J. Algebra 591 (2022), 217–248. [ IK25 ] Ishibashi, T . and S. Kano. “Unbounded sl 3 -laminations around punctures”. In: Math. Z. 310.4 (2025), P aper No. 88, 53 pages. [ Jon85 ] Jones, V . F . R. “A polynomial invariant for knots via von Neumann algebras”. In: Bull. Amer . Math. Soc. (N.S.) 12.1 (1985), 103–111. [ KP25 ] Karp, S. N. and M. E. Precup. Richardson tableaux and components of Springer fibers equal to Richardson varieties . Preprint, available at https: // arxiv .org / abs / 2506.20792 . 2025. [ Kho02 ] Khovanov , M. “A functor-valued invariant of tangles”. In: Algebr . Geom. T opol. 2 (2002), 665–741. [ Kho04a ] Khovanov , M. “Crossingless matchings and the cohomology of ( n , n ) Springer varieties”. In: Commmun. Contemp. Math 6.4 (2004), 561–577. [ Kho04b ] Khovanov , M. “sl(3) link homology”. In: Algebr . Geom. T opol. 4 (2004), 1045–1081. [ Kim24 ] Kim, J. Rotation invariant webs for three row flamingo Specht modules . Preprint, available at https: // arxiv .org / abs / 2402.11994 . 2024. [ KMY09 ] Knutson, A., E. Miller, and A. Y ong. “Gröbner geometry of vertex decompositions and of flagged tableaux”. In: J. Reine Angew . Math. 630 (2009), 1–31. [ Kos61 ] K ostant, B. “Lie algebra cohomology and the generalized Borel-Weil theorem”. In: Ann. of Math. (2) 74 (1961), 329–387. [ Kos63a ] Kostant, B. “Lie algebra cohomology and generalized Schubert cells”. In: Ann. of Math. (2) 77 (1963), 72–144. [ Kos63b ] Kostant, B. “Lie group representations on polynomial rings”. In: Amer . J. Math. 85 (1963), 327–404. [ KR84 ] K ung, J. P . S. and G.-C. R ota. “The invariant theory of binary forms”. In: Bull. Amer . Math. Soc. (N.S.) 10.1 (1984), 27–85. [ Kup96 ] Kuperberg, G. “Spiders for rank 2 Lie algebras”. In: Comm. Math. Phys. 180.1 (1996), 109– 151. [ Lam15 ] Lam, T . “Dimers, webs, and positroids”. In: J. Lond. Math. Soc. (2) 92.3 (2015), 633–656. [ LS82 ] Lascoux, A. and M.-P . Schützenberger. “P olynômes de Schubert”. In: C. R . Acad. Sci. P aris Sér . I Math. 294.13 (1982), 447–450. 26 [ LS24 ] Lê, T . T . Q. and A. S. Sikora. “Stated SL ( n ) -skein modules and algebras”. In: J. T opol. 17.3 (2024), P aper No. e12350, 93 pages. [ Lus81 ] Lusztig, G. “Green polynomials and singularities of unipotent classes”. In: Adv . Math. 42.2 (1981), 169–178. [ Man25 ] Mansour , R. “Counting two-column Young tableaux corresponding to smooth components of Springer fibers”. In: J. Algebraic Combin. 61.18 (2025), 9 pages. [ MV22 ] Mirkovi ´ c, I. and M. V ybornov. “Comparison of quiver varieties, loop Grassmannians and nilpotent cones in type A ”. In: Adv . Math. 407 (2022). W ith an appendix by V asily Krylov, P aper No. 108397, 54 pages. [ Nev24 ] Nevins, M. “The local character expansion as branching rules: nilpotent cones and the case of S L ( 2 ) ”. In: P acific J. Math 329.2 (2024), 259–301. [ OEIS ] OEIS F oundation Inc., The On-Line Encyclopedia of Integer Sequences . Published electroni- cally at https: // oeis.org . 2025. [ PP23 ] P atrias, R. and O . P echenik. “T ableau evacuation and webs”. In: Proc. Amer . Math. Soc. Ser . B 10 (2023), 341–352. [ PPS22 ] P atrias, R., O . P echenik, and J. Striker. “A web basis of invariant polynomials from non- crossing partitions”. In: Adv . Math. 408 (2022), P aper No. 108603, 33 pages. [ PS12 ] P errin, N. and E. Smirnov. “Springer fiber components in the two columns case for types A and D are normal”. In: Bull. Soc. Math. France 140.3 (2012), 309–333. [ PPR09 ] P etersen, T . K., P . Pylyavskyy, and B. Rhoades. “Promotion and cyclic sieving via webs”. In: J. Algebraic Combin. 30.1 (2009), 19–41. [ P os06 ] P ostnikov , A. “T otal positivity , Grassmannians, and networks”. Unpublished, available at arXiv:0609764 . (2006), 79 pages. [ PR21 ] Precup, M. and E. Richmond. “An equivariant basis for the cohomology of Springer fibers”. In: T rans. Amer . Math. Soc. Ser . B 8 (2021), 481–509. [ R TW32 ] R umer , Y ., E. T eller, and H. W eyl. “Eine für die V alenztheorie geeignette Basis der binären V ektorinvarianten”. In: Nachr . Ges. Wiss. gottingen Math.-Phys. Kl. (1932), 499–504. [ R us11 ] Russell, H. M. “A topological construction for all two-row Springer varieties”. In: P acific J. Math 253.1 (2011), 221–255. [ R T11 ] Russell, H. M. and J. S. T ymoczko. “Springer representations on the Khovanov Springer varieties”. In: Math. Proc. Cambridge Philos. Soc. 151.1 (2011), 59–81. [ Sag20 ] Sagan, B. E. Combinatorics: The Art of Counting . V ol. 210. Graduate Studies in Mathematics. American Mathematical Society , Providence, RI, 2020. [ SS06 ] Seidel, P . and I. Smith. “A link invariant of the symplectic geometry of nilpotent slices”. In: Duke Math. J. 134.3 (2006), 453–514. [ SSW25 ] Shen, L., Z. Sun, and D. W eng. “Intersections of dual SL 3 -webs”. In: T rans. Amer . Math. Soc. 378.8 (2025), 5513–5549. [ Slo80 ] Slodowy , P . F our lectures on simple groups and singularities . V ol. 11. Communications of the Mathematical Institute, Rijksuniversiteit Utrecht. 1980. [ Spa76 ] Spaltenstein, N. “The fixed point set of a unipotent transformation on the flag manifold”. In: Indag. Math. 38.5 (1976), 452–456. [ Spa82 ] Spaltenstein, N. Classes unipotentes et sous-groupes de Borel . V ol. 946. Lecture Notes in Math- ematics. Springer- V erlag, Berlin-New Y ork, 1982. [ Spr69 ] Springer , T . A. “The unipotent variety of a semi-simple group”. In: Algebraic Geometry (In- ternat. Colloq., Tata Inst. Fund. Res., Bombay , 1968) . V ol. 4. T ata Inst. Fund. Res., Bombay . 1969, 373–391. [ Spr76 ] Springer , T . A. “T rigonometric sums, Green functions of finite groups and representations of W eyl groups”. In: Invent. Math. 36 (1976), 173–207. [ Sta15 ] Stanley , R. P . Catalan Numbers . Cambridge University Press, New Y ork, 2015. 27 [ Ste96 ] Stembridge, J. R. “On the fully commutative elements of Coxeter groups”. In: J. Algebraic Combin. 5.4 (1996), 353–385. [ SW12 ] Stroppel, C. and B. W ebster. “2-block Springer fibers: convolution algebras and coherent sheaves”. In: Comment. Math. Helv . 87.2 (2012), 477–520. [ TL71 ] T emperley , H. N. V . and E. H. Lieb. “R elations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem”. In: Proc. Roy . Soc. London Ser . A 322.1549 (1971), 251–280. [ T ym17 ] T ymoczko, J. “The geometry and combinatorics of Springer fibers”. In: Around Langlands correspondences . V ol. 691. Contemp. Math. Amer . Math. Soc., Providence, RI, 2017, 359– 376. [ T ym25 ] T ymoczko, J. Quantum representations and webs . T alk at ICERM W orkshop on Diagrammatic Categorification, https: // icerm.brown.edu / video_archive / 4392 . 2025. [ vLee00 ] van Leeuwan, M. A. A. “Flag varieties and interpretations of Young tableau algorithms”. In: J. Algebra 224.2 (2000), 397–426. [ V ar79 ] V argas, J. A. “Fixed points under the action of unipotent elements of S L n in the flag variety”. In: Bol. Soc. Mat. Mex. 24 (1979), 1–14. [ W eh09 ] W ehrli, S. M. “A remark on the topology of ( n , n ) Springer varieties”. Unpublished, available at . (2009), 8 pages. [ Whi07 ] White, D. P ersonal communication to Brendon Rhoades. 2007. D E P T . O F C O M B I N A T O R I C S A N D O P T I M I Z A T I O N , U N I V E R S I T Y O F W A T E R L O O , W A T E R L O O ON N2L 3G1, C A N A DA Email address : mike.cummings@uwaterloo.ca